Find the sum of the series (-1)^n (5^(2n))/n!

1. sum = e^25
2. sum = e^−5
3. sum = 5
4. sum = e^5
5. sum = 25
6. sum = e^−25

25

To find the sum of the series (-1)^n (5^(2n))/n!, we can use the concept of the power series and the exponential function.

The power series representation of the exponential function is:

e^x = 1 + x/1! + x^2/2! + x^3/3! + ...

Comparing this to the given series, we can see that the x in the power series is 5^2 = 25.

Therefore, substituting x = 25 into the power series, we get:

e^25 = 1 + 25/1! + (25^2)/2! + (25^3)/3! + ...

Now let's rearrange the terms in the given series to match the form of the power series:

(-1)^n (5^(2n))/n! = 1 - 25/1! + (25^2)/2! - (25^3)/3! + ...

Comparing this to the power series for e^25, we can conclude that the sum of the given series is e^25, which corresponds to option 1.