Question - Power/taylor/MacLaurin series

**Sprintz** · November 9th 2009, 09:43 PM

Hello all; a homework question that is confusing me:

1) Find the Maclaurin Series for this function: f[x] = e^x + 2e^-x. Can I just make this

Sum(0 to Infinity) of x^n/(n!) + 2Sum(0 to Infinity) of (-x)^n/(n!) ??

**redsoxfan325** · November 9th 2009, 10:25 PM

Originally Posted by Sprintz

Hello all; a homework question that is confusing me:

1) Find the Maclaurin Series for this function: f[x] = e^x + 2e^-x. Can I just make this

Sum(0 to Infinity) of x^n/(n!) + 2Sum(0 to Infinity) of (-x)^n/(n!) ??

Yes, but you can combine some terms:

$\sum_{n=0}^{\infty}\frac{x^n}{n!}+2\sum_{n=0}^{\in fty}\frac{(-x)^n}{n!} = 1+2+x-2x+\frac{x^2}{2}+2\frac{x^2}{2}+\frac{x^3}{6}-2\frac{x^3}{6}+...= 3-x+\frac{3x^2}{2}-\frac{x^3}{6}+...$

If $n$ is even, the coefficient is $3$ ; if $n$ is odd, $-1$ ; it can therefore be represented as $1+2(-1)^n$

So the series is $\sum_{n=0}^{\infty}\frac{(1+2(-1)^n)x^n}{n!}$

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