Final answer:
The series 1/49+1/64+1/81+... is a convergent series with a sum of 1/42.
Step-by-step explanation:
The series 1/49+1/64+1/81+... is not divergent. In fact, it is a convergent series. To determine if a series is convergent or divergent, we need to look at the limit of its terms as n approaches infinity. In this case, the terms of the series approach zero as n becomes larger and larger. This means that the series converges to a finite sum.
To find the sum of the series, we can use the formula for the sum of an infinite geometric series. The formula is:
S = a/(1 - r), where a is the first term of the series and r is the common ratio (in this case, 1/7).
Substituting the values into the formula, we get:
S = (1/49)/(1 - 1/7) = 1/49 * 7/6 = 1/42.
Therefore, the sum of the series 1/49+1/64+1/81+... is 1/42.