Answers - Bruce E. Shapiro

Solutions
5. Find a power series for f(x) = x2
. What are its radius and interval of convergence?
1 + x
Using the formula for the sum of a geometric series, the power series is
x 2
1 + x = x2 ( 1 − x + x 2 − x 3 + · · · ) = x 2 − x 3 + x 4 − x 5 + · · ·
The formula for the geometric series holds for |x| < 1. Hence the radius of convergence
is 1 and the interval of convergence is −1 < x < 1.
6. How many terms of the series
∞∑ (−1) n+1
n=1
accuracy of of better than 0.00001?
n 5
This is an alternating series so we require
|a n+1 | =
are needed to compute the sum to an
1
(n + 1) 5 < 0.00001 = 10−5 = 1
10 5 =⇒ (n + 1)5 > 10 5 =⇒ n + 1 > 10 =⇒ n > 9
7. Find the radius and interval of convergence of the power series
∞∑
(−1) n xn
n 2 5 . n
Use the ratio test:
∣ ∣ ∣ a n+1 ∣∣∣ ∣ =
x n+1
a n
∣(n + 1) 2 5 × n2 5 n ∣∣∣ =
n 2 x ∣∣∣ n+1 x n ∣ → |x|
5(n + 1) 2 5 as n → ∞
We need this limit to be < 1 or |x| < 5 =⇒ −5 < x < 5. The RADIUS OF
CONVERGENCE is 5.
∞∑
Checking the endpoints: at x = −5, the sum is (−1) n (−5)n
n 2 5 = ∑ ∞
1
which is a
n n 2
n=1
n=1
p-series with p=2, so it CONVERGES.
∞∑
At x = 5, the series is (−1) n (5)n
n 2 5 = ∑ ∞
(−1) n
which CONVERGES ABSOLUTELY
n n 2
n=1
n=1
(because the series in the previous sentence converges) hence it CONVERGES.
Thus the INTERVAL of CONVERGENCE is −5 ≤ x ≤ 5.
8. Write the repeating decimal 4.17326326326 as a fraction of integers.
x = 4.17326326
1000x = 4173.26326326
999x = 4169.09
x = 4169.09
999
= 416909
99900
n=1
2