Problem 2: convergence.

2 . Determine whether the following converge or diverge:

a)

> sum( (k+3)! / (3*k)!, k=1..infinity);

Well, that is certainly interesting, but it is not too useful for most of you! By the ratio test,

> ( ((k+3+1))! /(3*(k+1))! ) / ((k+3)! / (3*k)! );

> simplify(%);

> limit(%, k=infinity);

Which is certainly smaller than 1. Therefore the series converges by the ratio test. Valid arguments can also be made with other comparisons.

b)

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

But are these terms small?

> limit( (-1)^k*(1-k)/(sqrt(k)),k = infinity);

Since this is not a finite number, the series diverges . by the basic divergence test 11.1.7.