Question

What is sum of the infinite geometric series for 1 -1/5 + 1/25 - 1/125 + ....

Answer 1

The sum of the given infinite geometric series 1 - 1/5 + 1/25 - 1/125 + ... is 5/6. This is found using the convergence formula for an infinite geometric series, S = a / (1 - r), with a = 1 and r = -1/5.

Step-by-step explanation:

Sum of the Infinite Geometric Series

The given series is an infinite geometric series with the first term a = 1 and a common ratio of r = -1/5. An infinite geometric series can be summed up using the formula S = a / (1 - r), provided that the absolute value of the common ratio is less than one (|r| < 1), which indicates that the series will converge. In this case, the series does converge because |r| = |-1/5| = 1/5, which is less than one.

To find the sum of the series, we apply the formula:

S = 1 / (1 - (-1/5)) = 1 / (1 + 1/5) = 1 / (6/5) = 5/6

Therefore, the sum of the infinite geometric series 1 - 1/5 + 1/25 - 1/125 + ... is 5/6.

Answer 2

5/6

Step-by-step explanation:

Sum of infinite geometric series is given by formula a/(1-r).

where a is the first term

and r is the common ratio of geometric term.

Common ratio for a geometric series is given by nth term/(n-1)th term.

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Given series is

1 -1/5 + 1/25 - 1/125 + ....

step 1 : calculate r

for r lets use second and first term

2nd term = -1/5

1st term = 1

r = 2nd term/1st term = (-1/5) / 1 = -1/5

sum of this series = a/(1-r)

substituting a with 1 and r with -1/5, we have

sum of this series = 1/(1-(-1/5)) = 1/(1+1/5)

=> sum of this series = 1/((5+1)/5)

=> sum of this series = 1/ (6/5) = 5/6

sum of the infinite geometric series 1 -1/5 + 1/25 - 1/125 + .... is 5/6