Determine whether the infinite geometric series converges. If so, find the sum

1/4+1/16+1/64+1/256....

The geometric sequence is

1/4 , 1/16 , 1/64 , 1/256 , ...
And the ratio is
r = 1/16 / (1/4) = 1/4

This infinite geometric sequence converges because the ratio is in the range of -1 < r < 1.

Getting the sum, we use the formula,
S = a1 / 1 - r
S = 1/4 / (1 - (1/4))
S = 1/4 / (3/4)
S = 1/3

Hope this helps~ `u`

Thank you!

To determine if the infinite geometric series converges, we need to check the common ratio (r) between consecutive terms. In this series, the common ratio (r) is the ratio between any term and its previous term.

In this case, each term is obtained by multiplying the previous term by 1/4. So, the common ratio (r) is 1/4.

For a geometric series to converge, the absolute value of the common ratio (|r|) must be less than 1.

In this case, |1/4| = 1/4 < 1.

Since the absolute value of the common ratio is less than 1, the geometric series will converge.

The formula to find the sum (S) of an infinite geometric series is given by:

S = a / (1 - r)

Where a is the first term of the series, and r is the common ratio.

In this case, the first term of the series (a) is 1/4, and the common ratio (r) is 1/4.

Substituting these values into the formula, we can find the sum (S):

S = (1/4) / (1 - 1/4)

S = (1/4) / (3/4)

S = (1/4) * (4/3)

S = 1/3

Therefore, the infinite geometric series 1/4 + 1/16 + 1/64 + 1/256 + ... converges to a sum of 1/3.

To determine if an infinite geometric series converges, we need to check if the common ratio (r) is between -1 and 1. In this case, our series is:

1/4 + 1/16 + 1/64 + 1/256 + ...

We can find the common ratio by dividing any term by the previous term. Let's divide the second term by the first term:

(1/16) / (1/4) = (1/16) * (4/1) = 1/64

Similarly, if we divide the third term by the second term, we get the same result. This pattern continues for all terms.

Since the common ratio (r) in this series is 1/64, which is between 0 and 1, we can conclude that the series converges.

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

Sum = a / (1 - r)

where 'a' is the first term and 'r' is the common ratio.

In our case:
a = 1/4 (the first term)
r = 1/64 (the common ratio)

Substituting these values into the formula, we have:

Sum = (1/4) / (1 - 1/64)

Simplifying further:

Sum = (1/4) / (63/64)
= (1/4) * (64/63)
= 16/63

Therefore, the sum of the infinite geometric series is 16/63.