Problem 2: convergence.

2 . Determine whether the following converge or diverge:

a)

> sum(sqrt(k)/(k^2 - 7), k=1..infinity);

Let's try the limit ratio test, with k^(-3/2):

> limit( (sqrt(k)/(k^2 - 7)) / (k^(-3/2)),k = infinity);

Since this is a finite number, and simce the sum of k^(-3/2) is a convergent p-series, series a) converges .

b)

> sum( (-1)^(k+1) * (2*k)! / (k+5)!, k=1..infinity);

Well, that is certainly interesting, but it is not convergent! By the ratio test,

> ( (2*(k+1))! /(k+1+5)! ) / ((2*k)! / (k+5)! );

> simplify(%);

> limit(%, k=infinity);

Which is certainly not smaller than 1. In fact, the terms a_k do not even tend to 0, so this series diverges by the basic divergence test, 11.1.7. (Or, if you will, by the ratio test. Valid arguments can also be made with other comparisons.)