Question

Find the exact sum of the infinite geometric series. If the series diverges, enter diverges. 4 - 1 + 1/4 - 1/16 + 1/64 - 1/256 ...

Answer 1

To find the exact sum of the series, we can use the formula for the sum of an infinite geometric series. The exact sum of the infinite geometric series 4 - 1 + 1/4 - 1/16 + 1/64 - 1/256 ... is 3.2.

Step-by-step explanation:

The given series is an infinite geometric series.

An infinite geometric series has the general form:
`a + ar + ar^2 + ar^3 + ...`

In this series, the first term 'a' is 4 and the common ratio 'r' is -1/4.

To find the exact sum of the series, we can use the formula for the sum of an infinite geometric series:

Sum = a / (1 - r)

Plugging in the values, we get:

Sum = 4 / (1 - (-1/4))

Simplifying, we get:

Sum = 4 / (1 + 1/4) = 4 / (5/4) = 16/5 = 3.2

Therefore, the exact sum of the infinite geometric series is 3.2.