Suppose is a convergent infinite geometric series

MATHEMATICS · Multi-part · A Level · Mathematics · Sequences and series · Sequences and series

Problem

Suppose

u_1 + u_2 + u_3 + u_4 + \dots

is a convergent infinite geometric series, with

u_1, u_2, u_3, u_4, \dots > 0

. (1) Explain why

u_1 - u_2 + u_3 - u_4 + \dots

and

\sqrt{u_1} + \sqrt{u_2} + \sqrt{u_3} + \sqrt{u_4} + \dots

must also be convergent infinite geometric series. (2) Given

u_1 - u_2 + u_3 - u_4 + \dots = \frac{81}{10}

and

\sqrt{u_1} + \sqrt{u_2} + \sqrt{u_3} + \sqrt{u_4} + \dots = \frac{9}{2}

, find

u_1 + u_2 + u_3 + u_4 + \dots