Write the first four nonzero terms and general term of the geometric series

Question

Write the first four nonzero terms and general term of the geometric series that is equal to 1/(1+t^5).

Answer 1

The sum of an infinite geometric series a + ar + ar² + ar³ + .... = a/(1 - r), where |r| < 1 or it won't converge.

You are given the series sum: a/(1 - r) = 1/(1 + t⁵)

That tells us a = 1, r = - t⁵.

Then, the series is 1 + (- t⁵) + ( - t⁵)² + ( - t⁵)³ + .... = 1 - t⁵ + t¹⁰ - t¹⁵ + ...

The general term is a_n = ar^n-1 = ( - t⁵)^n-1 = (-1)^n-1 t^5n-5 , for n = 1, 2, 3, ...