# Need help with the exact sum of convergent series

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• Nov 6th 2012, 04:46 AM
kimsanders82
Need help with the exact sum of convergent series
Please help with the following:

Find the exact sum of the following convergent series:

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

Thanks!
• Nov 6th 2012, 05:19 AM
Prove It
Re: Need help with the exact sum of convergent series
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?
• Nov 6th 2012, 06:13 AM
kimsanders82
Re: Need help with the exact sum of convergent series
Is it e^(2/3)?
• Nov 6th 2012, 03:21 PM
Prove It
Re: Need help with the exact sum of convergent series
Quote:

Originally Posted by kimsanders82
Is it e^(2/3)?

Yes, well done :)