2—Infinite Series 44Exercises1 Evaluate by handcos 0. 1 to four places.
2 In the same way, evaluatetan 0. 1 to four places.
3 Use the first two terms of the binomial expansion to estimate
√
2 =
√
1 + 1. What is the relative
error? [(wrong−right)/right]
4 Same as the preceding exercise, but for
√
1. 2.
5 What is the domain of convergence forx−x^4 +x^9 −x^4
2+x^5
2
−···6 Does
∑∞
n=0cos(n)−cos(n+ 1)converge?
7 Does
∑∞
n=11
√
n
converge?8 Does
∑∞
n=1n!
n^2
converge?9 What is the domain of convergence for
x
1. 2
−
x^2
2. 22
+
x^3
3. 33
−
x^4
4. 44
+···?
10 From Eq. (2.1), find a series for
1
(1−x)^2
.
11 Ifxis positive, sum the series1 +e−x+e−^2 x+e−^3 x+···
12 What is the ratio of the exact value of20!to Stirling’s approximation for it?
13 For the example in Eq. (2.22), what are the approximate values that would be predicted from
Eq. (2.26)?
14 Do the algebra to evaluate Eq. (2.25).
15 Translate this into a question about infinite series and evaluate the two repeating decimal numbers:
0. 444444 ..., 0. 987987987 ...
16 What does the integral test tell you about the convergence of the infinite series
∑∞
1 n
−p?17 What would the power series expansion for the sine look like if you require it to be valid in arbitrary