Sum of the convergent infinite series ln(n)/n^2

• mathgurl20
In summary, the problem is to find the sum of the series ln(n)/n^2 from n=1 to infinity, which has been determined to be convergent. The attempt at a solution involved trying to use geometric series and telescoping, but ultimately the solution was found using Stolz theorem.
mathgurl20

Homework Statement

Find the sum of the series: ln(n)/n^2 from n=1 to infinity.
I already know that it is convergent(at least i hope i am right on that fact)

The Attempt at a Solution

I tried to use geometric series but i can't see anything like that that would work, and i can't see a way to use telescoping. And just starting with n=1 and summing numbers didn't seem to get me anywhere either.

You still can use telescoping.
ln((n+1)^(1/(n+1)^2))-ln(n^(1/n^2)), use stolz theorem on this limit to get your answer, btw I am sure you know that but stolz theorem resembles L'hopital theorem.

Thank you very much :)

What is the formula for finding the sum of the convergent infinite series ln(n)/n^2?

The formula for finding the sum of the convergent infinite series ln(n)/n^2 is: S = ln(2π)/4 - 1/2

How do you know if the infinite series ln(n)/n^2 is convergent?

The infinite series ln(n)/n^2 is convergent if the limit of the sequence of partial sums approaches a finite number as n approaches infinity.

Can the sum of the convergent infinite series ln(n)/n^2 be calculated without using a formula?

Yes, the sum of the convergent infinite series ln(n)/n^2 can be calculated using summation techniques such as integration by parts or by using a computer program.

Is the sum of the convergent infinite series ln(n)/n^2 a rational or irrational number?

The sum of the convergent infinite series ln(n)/n^2 is an irrational number.

What are some real-world applications of the sum of the convergent infinite series ln(n)/n^2?

The sum of the convergent infinite series ln(n)/n^2 has various applications in mathematics, physics, and engineering, such as in calculating the area under certain curves, approximating integrals, and solving differential equations.

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