+0

# Algebra Question is very hard

0
56
2

Let , $$|r| < 1$$ ,

$$S = \sum^{\infty}_{k=0}r^k$$

and

$$T = \sum^{\infty}_{k=0}kr^k$$

Our approach is to write T as a geometric series in terms of S and r. Give a closed-form expression for T in terms of r

Jan 7, 2021

#1
0

Let f(r) = 1 + r + r^2 + ... = 1/(1 - r).

Taking the derivative: f'(r) = 1 + 2r + 3r^2 + ... = 2r/(1 - r)^2.

Then r + 2r^2 + 3r^3 + ... = 2r^2/(1 - r)^2.

Therefore, T = 2r^2/(1 - r)^2.

Jan 7, 2021
#2
+31514
+1

Like so:

Jan 7, 2021