Need help with the exact sum of convergent series

**kimsanders82** · November 6th 2012, 04:46 AM

Please help with the following:

Find the exact sum of the following convergent series:

sum from n=0 to infinity of (2ⁿ/(3ⁿn!))

Thanks!

**Prove It** · November 6th 2012, 05:19 AM

You should know that $\displaystyle \begin{align*} e^x = \sum_{n = 0}^{\infty} \frac{x^n}{n!} \end{align*}$ .

Notice that $\displaystyle \begin{align*} \frac{2^n}{3^n \, n!} = \frac{2^n}{3^n} \cdot \frac{1}{n!} = \left( \frac{2}{3} \right)^n \cdot \frac{1}{n!} = \frac{\left( \frac{2}{3} \right)^n }{n!} \end{align*}$ .

What do you think your sum has to equal?

**kimsanders82** · November 6th 2012, 06:13 AM

Is it e^(2/3)?

**Prove It** · November 6th 2012, 03:21 PM

Originally Posted by kimsanders82

Is it e^(2/3)?

Yes, well done

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