Question

Find the sum of the telescoping series.
∑(from k=1 to [infinity])[(1/8^(k+1) - 1/8^k)]

Answer 1

The sum of the telescoping series is 9/64.

Step-by-step explanation:

The given series is a telescoping series because most of the terms cancel each other out, leaving only a few terms at the end.

To find the sum of the series, we need to understand the pattern of the terms. We can start by simplifying the expression in each term:

1/8^(k+1) - 1/8^k = 1/8 * (1/8^k - 1) = 1/8 * (1 - 1/8^k)

Now, we can write out the first few terms of the series:

1/8 * (1 - 1/8) + 1/8 * (1 - 1/8^2) + 1/8 * (1 - 1/8^3) + ...

We can see that each term has the form 1/8 * (1 - 1/8^k). Notice that the (1 - 1/8) terms cancel out, leaving only the 1 term at the beginning and the -1/8^k terms at the end:

Sum of the series = 1/8 + (-1/8^2) + (-1/8^3) + ...= 1/8 + (-1/8)^2/(1 + 1/8) = 1/8 + 1/64/(9/8) = 1/8 + 1/512/9 = 9/64