Question

could someone help me to find the sum of the series from zero to infinity

(3^n/(5^n)n!)

Answer 1

If I understand your question correctly, you would like to know what is:\[\sum_{n=0}^{\infty} \frac{ 3^n }{ 5^n \times n! }\]

If this is the case, then, the equation can be written as:
\[\sum_{n=0}^{\infty} \frac{ (\frac{3}{5})^n }{n! }\]

The Taylor series of e^x is:
\[e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}\]

Comparing the two equations, the answer would be e^(3/5) = 1.8221.

Hope it helps!