Use the comparison test to show that the series 1 X n=1 p 1 n2+ 1 diverges also

The University of Waikato Department of Mathematics

Elements of Analysis math252-08B Tutorial/Workshop 3, 30th July 2008

Name:||||||||||-

Hand in this sheet at the end of the Workshop. Your name will be noted, but the working will not be assessed. Use both sides of this sheet and include at least some working on it.

1. Find the n'th partial sum of the series X1

n=0

(1 4)ⁿ

and then nd the sum of the series. Apply the ratio test to verify that it does converge. Then do the same for

X1 n=0

2^{n 1} 1 4ⁿ without nding the sum.

2. Verify that the seriesP₁

n=12n 1

n+1 diverges by applying the diver- gence test.

3. Assume the seriesP₁

n=11

n diverges. Use the comparison test to show that the series ₁

X

n=1

p 1 n²+ 1 diverges also. Then by comparison withP₁

n=1 1

n² =₆² show that X1

n=0

n + 2ⁿ n²2ⁿ converges.

1