# A few problems encountered in exam revision

• Jan 19th 2010, 11:44 AM
schteve
A few problems encountered in exam revision
Hi. Any help with these would be much appreciated.

1. By considering the infinite series (SUMMATION SIGN with infinity on top and n=0 on bottom) of x^n for x< 1 show that:

(SUMMATION SIGN with infinity on top and n=0 on bottom) (n^2)(x^n) = x(1-x^2)/(1-x)^4.

I know this is to do with differentiating the previous derivative (x/(1-x)^2), but i got in a mess with it.

2.Find the indefinate integral of x^2*e^(x^3)

This is integration by parts but the x^3 is confusing me as i don't know how to deal with it. Basically my question is; how do you integrate e^(x^3)?

3. Use your calculator to find the value of the following expression when x=0.01 to 3dp:

y(x) = (e^(2x) - 2(1 +2x)^(1/2) +1) / (cos (x/2) -1)

The previous part of the question asked me to obtain Maclaurin series and re-write the above expression which i did. However when i substituted x=0.01 the 2 answers from the series and the exact did not match. I maybe approaching this question incorrectly. What is the meothod for this

Again any help would be much appreciated.
• Jan 19th 2010, 12:02 PM
running-gag
Quote:

Originally Posted by schteve
Hi. Any help with these would be much appreciated.

1. By considering the infinite series (SUMMATION SIGN with infinity on top and n=0 on bottom) of x^n for x< 1 show that:

(SUMMATION SIGN with infinity on top and n=0 on bottom) (n^2)(x^n) = x(1-x^2)/(1-x)^4.

I know this is to do with differentiating the previous derivative (x/(1-x)^2), but i got in a mess with it.

Hi

This is what I would do

$\sum_{n=0}^{+\infty} x^n = \frac{1}{1-x}$

Differentiating with respect to x

$\sum_{n=0}^{+\infty} (n+1) x^n = \frac{1}{(1-x)^2}$

Differentiating again with respect to x

$\sum_{n=0}^{+\infty} (n+2)(n+1) x^n = \frac{2}{(1-x)^3}$

Using $(n+2)(n+1) = n^2 + 3n + 2 = n^2 + 3(n+1) - 1$ or $n^2 = (n+2)(n+1) - 3(n+1) + 1$ you are able to find

$\sum_{n=0}^{+\infty} n^2 x^n$
• Jan 19th 2010, 12:12 PM
General
Quote:

Originally Posted by schteve
2.Find the indefinate integral of x^2*e^(x^3)

This is integration by parts but the x^3 is confusing me as i don't know how to deal with it. Basically my question is; how do you integrate e^(x^3)?
.

No need for integration by parts.
the substitution $u=x^3$ will solve it easily :) .
By the way, It is not easy to evaluate $\int e^{(x^3)} dx$
since it has unelementary functions.
see this:
integrate e^(x^3) - Wolfram|Alpha