Point of convergence of a series

In summary, the problem is to find the sum $\displaystyle\sum_{k=1}^\infty \frac{k^2}{k!}$ and using the ratio test, we know that the series converges. To find what it converges to, we can rewrite it as $\displaystyle \sum_{k=1}^{\infty} \frac{k^{2}}{k!} = \sum_{k=1}^{\infty} \frac{k}{(k-1)!} = \sum_{k=0}^{\infty} \frac{k+1}{k!} = \sum_{k=0}^{\infty} \frac{k}{k!}
  • #1
caffeinemachine
Gold Member
MHB
816
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Hello MHB.

I have been preparing for my subject GRE and I need help on the following problem.

Find $\displaystyle\sum_{k=1}^\infty \frac{k^2}{k!}$.

Using the ratio test we know that the series converges but how to we find what it converges to?
 
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  • #2
caffeinemachine said:
Hello MHB.

I have been preparing for my subject GRE and I need help on the following problem.

Find $\displaystyle\sum_{k=1}^\infty \frac{k^2}{k!}$.

Using the ratio test we know that the series converges but how to we find what it converges to?

Is...

$\displaystyle \sum_{k=1}^{\infty} \frac{k^{2}}{k!} = \sum_{k=1}^{\infty} \frac{k}{(k-1)!} = \sum_{k=0}^{\infty} \frac{k+1}{k!} = \sum_{k=0}^{\infty} \frac{k}{k!} + \sum_{k=0}^{\infty} \frac{1}{k!} = 2\ e$

Kind regards

$\chi$ $\sigma$
 
  • #3
chisigma said:
Is...

$\displaystyle \sum_{k=1}^{\infty} \frac{k^{2}}{k!} = \sum_{k=1}^{\infty} \frac{k}{(k-1)!} = \sum_{k=0}^{\infty} \frac{k+1}{k!} = \sum_{k=0}^{\infty} \frac{k}{k!} + \sum_{k=0}^{\infty} \frac{1}{k!} = 2\ e$

Kind regards

$\chi$ $\sigma$
Taught me a lot. Thanks. :)
 

FAQ: Point of convergence of a series

What is the point of convergence of a series?

The point of convergence of a series is the value that the series approaches as the number of terms increases towards infinity. It is the limit of the sequence of partial sums of the series.

How is the point of convergence of a series determined?

The point of convergence can be determined by evaluating the series using various convergence tests, such as the ratio test or the root test. If the series converges, the limit of the sequence of partial sums will be the point of convergence.

What happens if the point of convergence of a series is not finite?

If the point of convergence is not finite, the series is said to diverge. This means that the series does not approach a specific value and does not have a point of convergence.

Can the point of convergence of a series be negative?

Yes, the point of convergence can be negative. It is simply the value that the series approaches, regardless of whether it is positive or negative.

How is the point of convergence of a series related to its convergence or divergence?

The point of convergence is directly related to the convergence or divergence of a series. If the series has a finite point of convergence, it is said to converge, whereas if the point of convergence is not finite, the series is said to diverge.

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