# How to prove this infinite series?

• B
• Imaxx
In summary, the conversation discussed an infinite series obtained while transforming the equation of the Basel problem. It was suggested that the series could be summed up using the telescope method, and it was proven that the series is equal to 2. However, the use of l'Hopital's rule was questioned and it was reminded that homework questions should not be fully answered and require effort from the original poster. The thread was then closed.
Imaxx
While transforming the equation of the Basel problem, the following infinite series was obtained.

$$\sum_{n=1}^{\infty} \frac{n^2+3n+1}{n^4+2n^3+n^2}=2$$

However I couldn't think of a simple way to prove that.
Can anyone prove that this equation holds true?

Your series can be put on the form of a telescope series and thereby summed up.
\begin{align*} \sum_{n=1}^\infty\frac{n^2+3n+1}{n^4+2n^3+n^2} &= \sum_{n=1}^\infty\frac{n^2+3n+1}{n^2(n+1)^2} \\ &= \lim_{N\rightarrow\infty}\sum_{n=1}^N\bigg\lbrace\frac{2n^2+3n}{(n+1)^2} - \frac{2(n-1)^2+3(n-1)}{n^2}\bigg\rbrace \\ &= \lim_{N\rightarrow\infty}\frac{2N^2+3N}{(N+1)^2} \\ &= 2, \end{align*}
where the last equality follows by successive use of l'Hopital's rule.

You don't need l'Hopital's rule here; just observe $$\frac{2N^2 + 3N}{(N + 1)^2} = \frac{N^2(2 + \frac 3N)}{N^2(1 + \frac1N)^2} = \frac{2 + \frac 3N}{(1 + \frac1N)^2}$$ and the result follows from the proposition that the limit of a ratio is the ratio of the limits provided the limit of the denominator is not zero.

This was a post in Calc & Beyond HW

I like to take the chance and remind you of our rules:
• do not provide full answers, that doesn't help the OP to understand their problem, even in technical forums
• do not open multiple threads on the same topic
• homework questions (anyway where they have been posted) require some efforts to be shown from the OP. We are not a solution automaton. Our goal is to teach, not to solve.

Imaxx

## 1. What is an infinite series?

An infinite series is a sum of an infinite number of terms, where each term is related to the previous one by a specific pattern or rule. It can be represented in the form of ∑(an), where 'n' is the term number and 'an' is the value of the nth term.

## 2. How can we prove the convergence of an infinite series?

One way to prove the convergence of an infinite series is by using the limit comparison test. This involves comparing the given series with a known convergent or divergent series and taking the limit of their ratio. If the limit is a finite non-zero number, then both series have the same convergence behavior.

## 3. What is the difference between absolute and conditional convergence?

Absolute convergence is when the series converges regardless of the order of the terms, while conditional convergence is when the series only converges when the terms are arranged in a specific order. In other words, absolute convergence guarantees convergence, while conditional convergence does not.

## 4. Can we use the ratio test to prove the convergence of all infinite series?

No, the ratio test can only be used for series with positive terms. Additionally, it is inconclusive for series with alternating signs or terms that do not approach zero as 'n' approaches infinity.

## 5. How can we prove the divergence of an infinite series?

One way to prove the divergence of an infinite series is by using the divergence test, where we take the limit of the terms of the series. If the limit is not equal to zero, then the series diverges. However, this test is inconclusive for series with alternating signs or terms that approach zero slowly.

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