Find the sum of each infinite series, or state that the sum doesn'nt exist

1/7+5/14+25/28+

(5/14) ÷ (1/7) = 2.5

(25/28)÷(5/14) = 2.5

From the first three terms I will assume that this is a geometric series with r = 2.5
since r > 1, it will diverge, thus it will not have a sum that can be calculated.
The sum will be infinite.

cucumbeer

To find the sum of an infinite series, we can examine the pattern and determine if it converges to a finite value or if it diverges (meaning it does not have a finite sum).

Let's look at the given series: 1/7 + 5/14 + 25/28 + ...

We can see that each term in the series is obtained by multiplying the previous term by 5/2.

To find the sum, we can use the formula for the sum of an infinite geometric series:
S = a / (1 - r)

In this formula, "a" represents the first term of the series, and "r" is the common ratio between consecutive terms.

In our given series, the first term "a" is 1/7, and the common ratio "r" is 5/2.

So, plugging these values into the formula:
S = (1/7) / (1 - 5/2)

To simplify this expression, we can multiply the numerator and denominator by 2:
S = 2/14 / (2/2 - 5/2)
S = 2/14 / (-3/2)
S = (2/14) * (-2/3)
S = -4/42

Simplifying further, we get:
S = -2/21

Therefore, the sum of the given infinite series 1/7 + 5/14 + 25/28 + ... is -2/21.