Question

By recognizing the series below as a Taylor series evaluated at a particular value of x, find the exact sum of the convergent series. 1 + 3/1! + 9/2! + 27/3! + 81/4! + ... + 3n/n! +.......

Answer 1

Answer:

$e^3$

Step-by-step explanation:

Given is a series as

$1+\frac{3}{1!} +\frac{3^2}{2!} +...+\frac{3^n}{n!} +...$

Recall the expansion of

$e^x = 1+x+\frac{x^2}{2!} +...+\frac{x^n}{n!} +...$

This expansion is valid for all real values of x.

Comparing this with our series we find that x =3

Hence the given series =$e^3$

Thus we find that the given series can be recognized with the expansion of exponential series with powers of e and here we see that power of e is 3.

So the given Taylor series is equivalent to

$e^3$