Anyone help me with these 2 calc convergence problems?

Question

Hey can anyone help me with these questions on my final exam review? It's the last thing we covered and we didn't have too much time to go over this stuff...

1.Find the interval of convergence for the series:
Summation(1 to infinity) of
(2^(k)) / ((2k)!) * x^(k)

2.Write the first five nonzero terms in a power series for
x / (2+3x^(3))
Find the interval of convergence. Use these five terms to estimate:
Integral from 0 to 0.1 (xdx) / (2+3x^(3))

Thanks y'all!!

Answer 1

In the first one you weren't too clear about whether it's (2/x) or 2x that's being taken to the kth power.

That said -- the real question is where the sum of N^k/(2k)! converges. Well, write M = N ^ (1/2), so that this is now the sum as j goes to infinity, even j only, of M^j/j! Once j gets to be bigger than 2M, each term is less than 1/2 of the one that preceded it. So you're always going to have convergence.

Number 2 is just a Taylor series question. Find the first terms, by taking lots of derivatives. Then integrate their sum to estimate the integral of the whole thing.

Answer 2

mathsmanretired's answer if truth be told makes use of the cut back try: the words strengthen and initiate beneficial, so as that they could't have cut back 0. Strictly speaking, the ratio try (as defined on the PlanetMath hyperlink under) is unquestionably inconclusive, because of the fact the limsup of the ratios is a million. quite often i'm responding because of the fact of your L'Hopital's Rule theory. you won't save the factorials consistent. you may opt to replace them with a sufficiently differentiable function which has a similar opinion with the factorial on the integers. The Gamma function (2nd hyperlink) is the function you may likely use, however this is needlessly overcomplicated while an basic ratio calculation suffices.