Series and Factorial: Solve the Sum of Infinite Terms

In summary, the conversation discussed using induction to prove the expression 1-\frac{1}{(n+1)(n!)} as the sum of the infinite series \sum_{k=1}^\infty \frac{k}{(k+1)!}. It was also mentioned that the expression can be derived analytically by considering it as a telescoping series or by using the derivative of \frac{e^x-1}{x}. Additionally, a new infinite series \sum_{n=1}^{\infty}\frac{8^{n}}{(n)!} was introduced and help was requested to solve it.
  • #1
thechunk
11
0
I’ve been playing around with the infinite series:
[tex] \sum_{k=1}^\infty \frac{k}{(k+1)!} [/tex]

I haven’t really gotten anywhere with it however I punched it into my calculator and it determined the sum to be 1. And the sum of n terms of the series equals
[tex]1-\frac{1}{(n+1)(n!)} [/tex]
Why is this so? Any help is much appreciated.
 
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  • #2
Use induction on that last statement. Show its true for n = 1, then assume it's true for n = k, and show it's true for n = k+1
 
  • #3
I see how I can use induction to find why [tex]1-\frac{1}{(n+1)(n!)} [/tex]
gives the sum of the series but how would you analytically come up with that expression in the first place. My calculator did it in a second, how did it generate the expression. Is there something I am missing?
 
  • #4
It's a telescoping series, this may help:

[tex]\sum_{k=1}^{n}\frac{k}{(k+1)!}=\sum_{k=1}^{n}\left(\frac{k+1}{(k+1)!}-\frac{1}{(k+1)!}\right)[/tex]


For the infinite series you can also consider:

[tex]\frac{d}{dx}\left(\frac{e^x-1}{x}\right)=\sum_{k=1}^{\infty}\frac{kx^{k-1}}{(k+1)!}[/tex]
 
  • #5
Thanks shmoe, I lost my negative and made the series, dare I say, even more infinite. Mwahahaha...
 
  • #6
Whay about : \sum_{n=1}^{\infty}\frac{8^{n}}{(n)!} ( I copy like this cause i don´t know how to put the symbol)Does anybody know how to solve this? PLease, help.
 

1. What is a series in mathematics?

A series is a mathematical concept that represents the sum of a sequence of terms. It can be finite or infinite, and each term in the series is added together to find the total sum.

2. How is a series represented in mathematical notation?

A series is typically represented using the sigma notation (∑), where the index of the series is listed below the sigma and the expression for the terms is listed above the sigma. For example, the series Σn=1 ∞ (1/n) represents the sum of the infinite terms 1/1 + 1/2 + 1/3 + ...

3. What is a factorial in mathematics?

A factorial is a mathematical operation denoted by an exclamation mark (!), where a non-negative integer is multiplied by all the positive integers less than it. For example, 4! (read as "four factorial") is equal to 4 x 3 x 2 x 1 = 24.

4. How is a factorial used in series?

A factorial is often used in series to represent the number of terms in the series. For example, the series Σn=1 4 (n!) represents the sum of the first 4 terms of the factorial sequence, which would be 1! + 2! + 3! + 4! = 33.

5. How do you solve the sum of infinite terms in a series using factorial?

In order to solve the sum of infinite terms in a series using factorial, you can use the formula ∑n=1 ∞ (1/n!) = e, where e is the mathematical constant approximately equal to 2.71828. This formula is derived from the Taylor series expansion of the exponential function e^x.

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