Thread: A few problems encountered in exam revision

1. 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.

2. 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

$\sum_{n=0}^{+\infty} n^2 x^n$

3. 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