How do you convert functions into partial sums?

Question

for example, how would you do this problem?

Consider the function ln( 1 + 6 x).
Write a partial sum for the power series which represents this function consisting of the first 5 nonzero terms. Also indicate the radius of convergence.

Answer 1

Let F be any differentiable function. Then you can write a McLauren series for F(x) in the neighborhood of 0 as:

F(x) = F(0) + F'(0)x + (1/2!)F''(0)x^2 + (1/3!)F'''(0)x^3 + ...

In this case F(x) is ln(1+6x), so to compute F'(x), etc. you need to use the chain rule:

if h(x) = f(g(x)) then
dh/dx = (df/du)(du/dx)

In this case:
f(u) = ln u
g(x) = 1 + 6x

This takes care of the first part of the question. (Don't forget to substitute (1+6x) for u when computing F', etc.)

But just because the infinite series exists, doesn't mean it always converges.

For example:

S(x) = 1 + x + (1/2)x^2 + (1/3)x^3 + (1/4)x^4 + ... + (1/n)x^n + ...

converges for all x such that |x| < 1. But for x = 1 we get

S(1) = 1 + 1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ...

which doesn't converge even though the terms get smaller.

And if |x| > 1, the terms eventually start getting bigger and bigger and so the sum can't converge.

So the radius of convergence of this series is 1.)

How do you tell if or when the sum of the terms in a series converges? Sometimes it can be hard, but in both these cases, it turns out to be straightforward. Check out:
http://en.wikipedia.org/wiki/Convergent_series

The series S fails the Cauchy condensation test at x = 1. (The test sequence becomes 1 + 1 + 1 + ... which clearly doesn't converge) but it passes the ratio test for all |x| < 1.

And the series for F(x) will have its radius of convergence determined by the alternating series test.