# Another sum of series quesiton

• ILoveBaseball
In summary, the series given is \sum_{n=1}^\infty \frac{n}{\sqrt{5n^2+5}} and there is no common ratio, indicating that it is not a geometric series. However, it can be simplified to \frac{1}{\sqrt{5}}\sum_{n=1}^\infty \frac{n}{\sqrt{(n^2+1)}}. The sequence a_{n}=\frac{n}{\sqrt{(n^2+1)}} does not converge to zero as n approaches infinity, which means that the series is divergent. This was proven mathematically by showing that the limit of a_{n} is not equal to zero.
ILoveBaseball
Consider the series

$$\sum_{n=1}^\infty \frac{n}{\sqrt{5n^2+5}}$$

Value ______

$$a_1 = .316227766, a_2 = 2/5, a_3 = .4242640687, a_4 = .4338609156$$

there doesn't seem to be any common ratio, so that means that this isn't a geometric series right?

well i think i can simplify the equation to:

$$\frac{1}{\sqrt{5}}\sum_{n=1}^\infty \frac{n}{\sqrt{(n^2+1)}}$$

hmm, that's as far as i got, can someone help me ?

Last edited:
Reevaluate $a_2$

i made the changes, but i still don't see a common ratio

$$\frac{1}{\sqrt{5}}\sum_{n=1}^\infty \frac{n}{\sqrt{(n^2+1)}}$$

That thing is only going to have a value if it's convergent, correct? I don't really remember much about this stuff.

If a series converges to a finite value, its sequence a must converge to zero.

Contrapositively, if the sequence a does not converge to zero, the series does not converge to a finite value.

Here, the sequence a is
$$\frac{n}{\sqrt{(n^2+1)}}$$,
which converges to unity, not zero, as n approaches infinity.
This implies that the series in question is divergent.

Right?

it converges for sure, it was the first question asked.

It does not converge ILoveBaseball; you must have mistyped.

If $$a_{n}=\frac{n}{\sqrt{5n^{2}+5}}$$
then,
$$\lim_{n\to\infty}a_{n}=\frac{1}{\sqrt{5}}>0$$
But a necessary requirement for convergence of the series $$\sum_{n=1}^{\infty}a_{n}$$ is that we have $$\lim_{n\to\infty}a_{n}=0$$

What reason do you have to think it does converge? Prove it mathematically.

sorry, you guys are right. for some reason there was a glitch, even if i selected converge, i got 50% of the problem correct(which means i got the first question right). but if i selected diverge, i got 100% of the problem(two problems = 100%) correct.

## 1. What is a sum of series question?

A sum of series question is a mathematical problem that involves finding the sum of a given series of numbers. This is usually done by adding up all the terms in the series.

## 2. How do I solve a sum of series question?

To solve a sum of series question, you need to identify the pattern or formula that the series follows. Once you have identified the pattern, you can use it to calculate the sum of the series. Alternatively, you can also use the formula for the sum of a finite arithmetic or geometric series depending on the type of series given.

## 3. What are the types of series that can be used in sum of series questions?

The most common types of series used in sum of series questions are arithmetic series, geometric series, and harmonic series. However, there are many other types of series that can be used, such as power series, Taylor series, and Fourier series.

## 4. Is there a shortcut or trick to solve sum of series questions quickly?

Yes, there are some shortcuts or tricks that can be used to solve sum of series questions quickly. For example, for arithmetic series, you can use the formula Sn = n/2(a1 + an), where Sn is the sum of the first n terms, a1 is the first term, and an is the last term. Similarly, for geometric series, you can use the formula Sn = a1(1-r^n)/(1-r), where r is the common ratio and n is the number of terms.

## 5. Can sum of series questions be used in real-world applications?

Yes, sum of series questions have many real-world applications in fields such as physics, engineering, and finance. For example, in physics, series are used to calculate the total distance traveled by an object with changing velocity. In finance, series are used to calculate compound interest and amortization. In engineering, series are used to model and analyze complex systems.

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