Calculating Infinite Series Sum: Methods for Convergence and Divergence

[dekoi]

The sum of a series:
$$\sum _{n=0} ^{\infty} \frac{2^{2n+1}x^{2n}}{n!}$$
is:
a)$$2cos(2x)$$
b)$$cos(x^2)$$
c)$$e^{2x}$$
d)$$2e^{2x^2}$$
e) None of the above.I have absolutely no idea how I would go about solving this. I know various tests for convergence and divergence, but the only methods I have to calculate sums are the geometric sum method, and the approximation method. I'm not sure how to solve this.

Any help is greatly appreciated, Thank You.

 

[StatusX]

Do you know about taylor series? Specifically, what is the taylor series for e^x?

 

[dekoi]

No I haven't learned the Taylor Series. Is that the only way of doing this question?

 

[Archon]

What's the sum of the series for x = 0? How does it compare to the values of the listed functions at
x = 0?

 

[HallsofIvy]

It also might help you to note that the sum is never negative.

 

[dekoi]

Does this require knowledge of Talor Series?

 

[VietDao29]

Does this require knowledge of Talor Series?

No, it does not.
As Archon has pointed out in post #4, if x = 0, then what's the value of the sum? Is there any of the listed functions which returns the same value as the sum of that series at x = 0?
Can you go from here?

 

[StatusX]

You might be able to use the process of elimination here, but that's not a very useful approach in general. And how could you eliminate the none of the above option?

The only other way I can think of is to find a differential equation this series solves and then find the analytic solution to it. That is, defining the series as a function y(x), find an equation relating y to y' and then solve for y(x).