Question 1

The limit of the sequence  an=en+3a _ { n } = e ^ { n + 3 }an​=en+3  is

Accepted Answer

A)  0
B)    75\frac { 7 } { 5 }57​ 
C)    57\frac { 5 } { 7 }75​ 
D)    −56- \frac { 5 } { 6 }−65​ 
E)   ∞\infty∞

Question 2

The sum of the series  ∑n=1∞π−n\sum _ { n = 1 } ^ { \infty } \pi ^ { - n }∑n=1∞​π−n  is

Accepted Answer

A)    π−1\pi - 1π−1 
B)    1π−1\frac { 1 } { \pi - 1 }π−11​ 
C)    π+1\pi + 1π+1 
D)    1π+1\frac { 1 } { \pi + 1 }π+11​ 
E)   ∞\infty∞

Question 3

The Taylor series representation of  f(x) =sin⁡xf ( x )  = \sin xf(x) =sinx  at  π4\frac { \pi } { 4 }4π​  is

Accepted Answer

A)    ∑n=0∞sin⁡(π(2n+1) 4) (x−π4) nn!\sum _ { n = 0 } ^ { \infty } \frac { \sin \left( \frac { \pi ( 2 n + 1 )  } { 4 } \right)  \left( x - \frac { \pi } { 4 } \right)  ^ { n } } { n ! }∑n=0∞​n!sin(4π(2n+1) ​) (x−4π​) n​ 
B)    ∑n=0∞sin⁡(π(2n+3) 4) (x−π4) nn!\sum _ { n = 0 } ^ { \infty } \frac { \sin \left( \frac { \pi ( 2 n + 3 )  } { 4 } \right)  \left( x - \frac { \pi } { 4 } \right)  ^ { n } } { n ! }∑n=0∞​n!sin(4π(2n+3) ​) (x−4π​) n​ 
C)    ∑n=0∞sin⁡(π(2n−1) 4) (x−π4) nn!\sum _ { n = 0 } ^ { \infty } \frac { \sin \left( \frac { \pi ( 2 n - 1 )  } { 4 } \right)  \left( x - \frac { \pi } { 4 } \right)  ^ { n } } { n ! }∑n=0∞​n!sin(4π(2n−1) ​) (x−4π​) n​ 
D)    ∑n=0∞sin⁡(π(2n−3) 4) (x−π4) nn!\sum _ { n = 0 } ^ { \infty } \frac { \sin \left( \frac { \pi ( 2 n - 3 )  } { 4 } \right)  \left( x - \frac { \pi } { 4 } \right)  ^ { n } } { n ! }∑n=0∞​n!sin(4π(2n−3) ​) (x−4π​) n​ 
E)    ∑n=0∞sin⁡(π(2n+5) 4) (x−π4) nn!\sum _ { n = 0 } ^ { \infty } \frac { \sin \left( \frac { \pi ( 2 n + 5 )  } { 4 } \right)  \left( x - \frac { \pi } { 4 } \right)  ^ { n } } { n ! }∑n=0∞​n!sin(4π(2n+5) ​) (x−4π​) n​

Question 4

The sum of the series  ∑n=1∞2n+35n\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { n + 3 } } { 5 ^ { n } }∑n=1∞​5n2n+3​  is

Accepted Answer

A)    163\frac { 16 } { 3 }316​ 
B)    83\frac { 8 } { 3 }38​ 
C)    43\frac { 4 } { 3 }34​ 
D)    32\frac { 3 } { 2 }23​ 
E)   ∞\infty∞

Question 5

Using the first three nonzero terms of a Maclaurin series, the approximated value of  ∫12e−x2xdx\int _ { 1 } ^ { 2 } \frac { e ^ { - x ^ { 2 } } } { x } d x∫12​xe−x2​dx  is

Accepted Answer

A)    3+4ln⁡22\frac { 3 + 4 \ln 2 } { 2 }23+4ln2​ 
B)    3+6ln⁡24\frac { 3 + 6 \ln 2 } { 4 }43+6ln2​ 
C)    3+8ln⁡26\frac { 3 + 8 \ln 2 } { 6 }63+8ln2​ 
D)    3+8ln⁡28\frac { 3 + 8 \ln 2 } { 8 }83+8ln2​ 
E)    3+6ln⁡212\frac { 3 + 6 \ln 2 } { 12 }123+6ln2​

Question 6

Using the first three nonzero terms of the Maclaurin series, the approximated value of  ∫12ex2x2dx\int _ { 1 } ^ { 2 } \frac { e ^ { x ^ { 2 } } } { x ^ { 2 } } d x∫12​x2ex2​dx  is

Accepted Answer

A)  3
B)    83\frac { 8 } { 3 }38​ 
C)    73\frac { 7 } { 3 }37​ 
D)    113\frac { 11 } { 3 }311​ 
E)    133\frac { 13 } { 3 }313​

Question 7

Which of the following is true for the infinite series  ∑n=1∞sin⁡(nπ) n?\sum _ { n = 1 } ^ { \infty } \frac { \sin ( n \pi )  } { n } ?∑n=1∞​nsin(nπ) ​?

Accepted Answer

A)  Its sequence of partial sums is not bounded from above.
B)  Its sequence of partial sums is positive.
C)  It is absolutely convergent.
D)  It is not conditionally convergent.
E)  It is divergent.

Question 8

The sum of the series  ∑n=2∞1n2−1\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n ^ { 2 } - 1 }∑n=2∞​n2−11​  is

Accepted Answer

A)  0
B)    12\frac { 1 } { 2 }21​ 
C)    34\frac { 3 } { 4 }43​ 
D)    34\frac { 3 } { 4 }43​ .
E)   ∞\infty∞

Question 9

The interval of convergence of the power series  ∑n=0∞(x+3) nn!\sum _ { n = 0 } ^ { \infty } \frac { ( x + 3 )  ^ { n } } { n ! }∑n=0∞​n!(x+3) n​  is

Accepted Answer

A)  (-4,-2) 
B)  [-4,-2) 
C)  (-4,-2]
D)  [-4,-2]
E)  (- ∞\infty∞ ,  ∞\infty∞  )

Question 10

The limit of the sequence  an=3n+2n+5a _ { n } = \frac { 3 n + 2 } { \sqrt { n } + 5 }an​=n​+53n+2​  is

Accepted Answer

A)  0
B)  3
C)    3\sqrt { 3 }3c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​ 
D)    25\frac { 2 } { 5 }52​ 
E)   ∞\infty∞

Question 11

Which of the following is true for the infinite series  ∑n=1∞cos⁡(n) n2?\sum _ { n = 1 } ^ { \infty } \frac { \cos ( n )  } { n ^ { 2 } } ?∑n=1∞​n2cos(n) ​?

Accepted Answer

A)  Its sequence of partial sums is not bounded from above.
B)  Its sequence of partial sums is decreasing.
C)    lim⁡n→∞cos⁡(n) n2≠0\lim _ { n ightarrow \infty } \frac { \cos ( n )  } { n ^ { 2 } } 
eq 0limn→∞​n2cos(n) ​=0 
D)  It is convergent.
E)  It is divergent.

Question 12

Which of the following is true for the infinite series  ∑n=1∞45n−1?\sum _ { n = 1 } ^ { \infty } \frac { 4 } { 5 ^ { n } - 1 } ?∑n=1∞​5n−14​?

Accepted Answer

A)  Its sequence of partial sums is not bounded from above.
B)  Its sequence of partial sums is decreasing.
C)    lim⁡n→∞45n−1≠0\lim _ { n ightarrow \infty } \frac { 4 } { 5 ^ { n } - 1 } 
eq 0limn→∞​5n−14​=0 
D)  It is convergent.
E)  It is divergent.

Question 13

Which of the following is true for the infinite series  ∑n=1∞2n+5(n2+5n) 3?\sum _ { n = 1 } ^ { \infty } \frac { 2 n + 5 } { \left( n ^ { 2 } + 5 n \right)  ^ { 3 } } ?∑n=1∞​(n2+5n) 32n+5​?

Accepted Answer

A)  Its sequence of partial sums is not bounded from below.
B)  Its sequence of partial sums is decreasing.
C)    lim⁡n→∞2n+5(n2+5n) 3≠0\lim _ { n ightarrow \infty } \frac { 2 n + 5 } { \left( n ^ { 2 } + 5 n ight)  ^ { 3 } } 
eq 0limn→∞​(n2+5n) 32n+5​=0 
D)  It is convergent.
E)  It is divergent.

Question 14

Which of the following is true for the infinite series  ∑n=1∞ln⁡(2n) en?\sum _ { n = 1 } ^ { \infty } \frac { \ln \left( 2 ^ { n } \right)  } { e ^ { n } } ?∑n=1∞​enln(2n) ​?

Accepted Answer

A)  Its sequence of partial sums is not bounded from above.
B)  Its sequence of partial sums is decreasing.
C)    lim⁡n→∞ln⁡2nen≠0\lim _ { n ightarrow \infty } \frac { \ln 2 ^ { n } } { e ^ { n } } 
eq 0limn→∞​enln2n​=0 
D)  It is convergent.
E)  It is divergent.

Question 15

Which of the following is true for the infinite series  ∑n=1∞(−π) n?\sum _ { n = 1 } ^ { \infty } ( - \pi )  ^ { n } ?∑n=1∞​(−π) n?

Accepted Answer

A)  Its sequence of partial sums is bounded from above.
B)  Its sequence of partial sums is decreasing.
C)  It is absolutely convergent.
D)  It is conditionally convergent.
E)  It is divergent.

Question 16

The limit of the sequence  an=3n+2n+5a _ { n } = \frac { 3 \sqrt { n } + 2 } { \sqrt { n } + 5 }an​=n​+53n​+2​  is

Accepted Answer

A)  0
B)  3
C)    3\sqrt { 3 }3c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​ 
D)    25\frac { 2 } { 5 }52​ 
E)   ∞\infty∞

Question 17

Which of the following is true for the infinite series  ∑n=1∞2−nn?\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { - n } } { \sqrt { n } } ?∑n=1∞​n​2−n​?

Accepted Answer

A)  Its sequence of partial sums is not bounded from above.
B)  Its sequence of partial sums is decreasing.
C)    lim⁡n→∞2−nn≠0\lim _ { n ightarrow \infty } \frac { 2 ^ { - n } } { \sqrt { n } } 
eq 0limn→∞​nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​2−n​=0 
D)  It is convergent.
E)  It is divergent.

Question 18

The sum of the series  ∑n=1∞π−n\sum _ { n = 1 } ^ { \infty } \pi ^ { - n }∑n=1∞​π−n  is

Accepted Answer

A)  0
B)  1
C)  -1
D)    12\frac { 1 } { 2 }21​ 
E)   ∞\infty∞

Question 19

The sum of the series  ∑n=1∞(2e) n\sum _ { n = 1 } ^ { \infty } \left( \frac { 2 } { e } \right)  ^ { n }∑n=1∞​(e2​) n  is

Accepted Answer

A)  0
B)    2e−2\frac { 2 } { e - 2 }e−22​ 
C)    22−e\frac { 2 } { 2 - e }2−e2​ 
D)    e−22\frac { e - 2 } { 2 }2e−2​ 
E)   ∞\infty∞

Question 20

By the error estimation theorem for convergent alternating series, the upper estimate to the error in approximating the sum of  ∑n=1∞(−1) nn\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 )  ^ { n } } { n }∑n=1∞​n(−1) n​  using the first three terms is

Accepted Answer

A)    12\frac { 1 } { 2 }21​ 
B)    14\frac { 1 } { 4 }41​ 
C)    18\frac { 1 } { 8 }81​ 
D)    116\frac { 1 } { 16 }161​ 
E)    132\frac { 1 } { 32 }321​

Exam 9: Infinite Series

The sum of the series ∑n=2∞1n2−1\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n ^ { 2 } - 1 }∑n=2∞​n2−11​ is

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Using the first three nonzero terms of a Maclaurin series, the approximated value of ∫12e−x2xdx\int _ { 1 } ^ { 2 } \frac { e ^ { - x ^ { 2 } } } { x } d x∫12​xe−x2​dx is

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The interval of convergence of the power series ∑n=0∞(x+3) nn!\sum _ { n = 0 } ^ { \infty } \frac { ( x + 3 ) ^ { n } } { n ! }∑n=0∞​n!(x+3) n​ is

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Which of the following is true for the infinite series ∑n=1∞ln⁡(2n) en?\sum _ { n = 1 } ^ { \infty } \frac { \ln \left( 2 ^ { n } \right) } { e ^ { n } } ?∑n=1∞​enln(2n) ​?

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Which of the following is true for the infinite series ∑n=1∞2n+5(n2+5n) 3?\sum _ { n = 1 } ^ { \infty } \frac { 2 n + 5 } { \left( n ^ { 2 } + 5 n \right) ^ { 3 } } ?∑n=1∞​(n2+5n) 32n+5​?

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The sum of the series ∑n=1∞(2e) n\sum _ { n = 1 } ^ { \infty } \left( \frac { 2 } { e } \right) ^ { n }∑n=1∞​(e2​) n is

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The limit of the sequence an=3n+2n+5a _ { n } = \frac { 3 \sqrt { n } + 2 } { \sqrt { n } + 5 }an​=n​+53n​+2​ is

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The Taylor series representation of f(x) =sin⁡xf ( x ) = \sin xf(x) =sinx at π4\frac { \pi } { 4 }4π​ is

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The sum of the series ∑n=1∞2n+35n\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { n + 3 } } { 5 ^ { n } }∑n=1∞​5n2n+3​ is

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The limit of the sequence an=en+3a _ { n } = e ^ { n + 3 }an​=en+3 is

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Which of the following is true for the infinite series ∑n=1∞2−nn?\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { - n } } { \sqrt { n } } ?∑n=1∞​n​2−n​?

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Which of the following is true for the infinite series ∑n=1∞cos⁡(n) n2?\sum _ { n = 1 } ^ { \infty } \frac { \cos ( n ) } { n ^ { 2 } } ?∑n=1∞​n2cos(n) ​?

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Which of the following is true for the infinite series ∑n=1∞(−π) n?\sum _ { n = 1 } ^ { \infty } ( - \pi ) ^ { n } ?∑n=1∞​(−π) n?

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Which of the following is true for the infinite series ∑n=1∞sin⁡(nπ) n?\sum _ { n = 1 } ^ { \infty } \frac { \sin ( n \pi ) } { n } ?∑n=1∞​nsin(nπ) ​?

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The sum of the series ∑n=1∞π−n\sum _ { n = 1 } ^ { \infty } \pi ^ { - n }∑n=1∞​π−n is

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The limit of the sequence an=3n+2n+5a _ { n } = \frac { 3 n + 2 } { \sqrt { n } + 5 }an​=n​+53n+2​ is

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By the error estimation theorem for convergent alternating series, the upper estimate to the error in approximating the sum of ∑n=1∞(−1) nn\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n }∑n=1∞​n(−1) n​ using the first three terms is

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Which of the following is true for the infinite series ∑n=1∞45n−1?\sum _ { n = 1 } ^ { \infty } \frac { 4 } { 5 ^ { n } - 1 } ?∑n=1∞​5n−14​?

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Using the first three nonzero terms of the Maclaurin series, the approximated value of ∫12ex2x2dx\int _ { 1 } ^ { 2 } \frac { e ^ { x ^ { 2 } } } { x ^ { 2 } } d x∫12​x2ex2​dx is

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The sum of the series ∑n=1∞π−n\sum _ { n = 1 } ^ { \infty } \pi ^ { - n }∑n=1∞​π−n is

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The sum of the series $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n ^ { 2 } - 1 }$ is

Correct Answer
verified

Using the first three nonzero terms of a Maclaurin series, the approximated value of $\int _ { 1 } ^ { 2 } \frac { e ^ { - x ^ { 2 } } } { x } d x$ is

Correct Answer
verified

The interval of convergence of the power series $\sum _ { n = 0 } ^ { \infty } \frac { ( x + 3 ) ^ { n } } { n ! }$ is

Correct Answer
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Which of the following is true for the infinite series $\sum _ { n = 1 } ^ { \infty } \frac { \ln \left( 2 ^ { n } \right) } { e ^ { n } } ?$

Correct Answer
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Which of the following is true for the infinite series $\sum _ { n = 1 } ^ { \infty } \frac { 2 n + 5 } { \left( n ^ { 2 } + 5 n \right) ^ { 3 } } ?$

Correct Answer
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The sum of the series $\sum _ { n = 1 } ^ { \infty } \left( \frac { 2 } { e } \right) ^ { n }$ is

Correct Answer
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The limit of the sequence $a _ { n } = \frac { 3 \sqrt { n } + 2 } { \sqrt { n } + 5 }$ is

Correct Answer
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The Taylor series representation of $f ( x ) = \sin x$ at $\frac { \pi } { 4 }$ is

Correct Answer
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The sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { n + 3 } } { 5 ^ { n } }$ is

Correct Answer
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The limit of the sequence $a _ { n } = e ^ { n + 3 }$ is

Correct Answer
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Which of the following is true for the infinite series $\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { - n } } { \sqrt { n } } ?$

Correct Answer
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Which of the following is true for the infinite series $\sum _ { n = 1 } ^ { \infty } \frac { \cos ( n ) } { n ^ { 2 } } ?$

Correct Answer
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Which of the following is true for the infinite series $\sum _ { n = 1 } ^ { \infty } ( - \pi ) ^ { n } ?$

Correct Answer
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Which of the following is true for the infinite series $\sum _ { n = 1 } ^ { \infty } \frac { \sin ( n \pi ) } { n } ?$

Correct Answer
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The sum of the series $\sum _ { n = 1 } ^ { \infty } \pi ^ { - n }$ is

Correct Answer
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The limit of the sequence $a _ { n } = \frac { 3 n + 2 } { \sqrt { n } + 5 }$ is

Correct Answer
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By the error estimation theorem for convergent alternating series, the upper estimate to the error in approximating the sum of $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n }$ using the first three terms is

Correct Answer
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Which of the following is true for the infinite series $\sum _ { n = 1 } ^ { \infty } \frac { 4 } { 5 ^ { n } - 1 } ?$

Correct Answer
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Using the first three nonzero terms of the Maclaurin series, the approximated value of $\int _ { 1 } ^ { 2 } \frac { e ^ { x ^ { 2 } } } { x ^ { 2 } } d x$ is

Correct Answer
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The sum of the series $\sum _ { n = 1 } ^ { \infty } \pi ^ { - n }$ is

Correct Answer
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