What Is the Sum of the Series \( \sum_{n=1}^\infty \frac{n}{(n+1)!} \)?

  • MHB
  • Thread starter hobomath
  • Start date
  • Tags
    Series Sum
In summary, the formula for finding the sum of a series is S = a + ar + ar^2 + ar^3 + ... + ar^n-1, where a is the first term, r is the common ratio, and n is the number of terms in the series. The number of terms in a series can be determined by counting the number of terms in the sequence or by using the formula n = (log S - log a) / log r, where S is the sum of the series, a is the first term, and r is the common ratio. The common ratio in a geometric series is the constant ratio between each term in the series and can be found by dividing any term by the previous term. The sum
  • #1
hobomath
4
0
Find the sum of this series:
$$ \sum_{n=1}^\infty \frac{n}{(n+1)!} $$

I'm really struggling with this one.. Any help will be highly appreciated. Thanks you.
 
Physics news on Phys.org
  • #2
Look at the sequence of partial sums.

\(\displaystyle S_1 = \dfrac{1}{2}\), \(\displaystyle S_2 = \dfrac{5}{6}\), \(\displaystyle S_3 = \dfrac{23}{24}\), \(\displaystyle S_4 = \dfrac{119}{120}\), \(\displaystyle S_5 = \dfrac{719}{720}\), etc.

Once you guess the obvious answer for \(\displaystyle S_n\), you can prove it by induction. Then take the limit as \(\displaystyle n \rightarrow \infty\)
 

Related to What Is the Sum of the Series \( \sum_{n=1}^\infty \frac{n}{(n+1)!} \)?

1. What is the formula for finding the sum of a series?

The formula for finding the sum of a series is S = a + ar + ar^2 + ar^3 + ... + ar^n-1, where a is the first term, r is the common ratio, and n is the number of terms in the series.

2. How do you determine the number of terms in a series?

The number of terms in a series can be determined by counting the number of terms in the sequence or by using the formula n = (log S - log a) / log r, where S is the sum of the series, a is the first term, and r is the common ratio.

3. What is the common ratio in a geometric series?

The common ratio in a geometric series is the constant ratio between each term in the series. It can be found by dividing any term by the previous term.

4. Can the sum of an infinite series be calculated?

Yes, the sum of an infinite series can be calculated if the common ratio is less than 1. The formula for calculating the sum of an infinite geometric series is S = a / (1 - r), where a is the first term and r is the common ratio.

5. What is the difference between a finite and infinite series?

A finite series has a limited number of terms, while an infinite series has an infinite number of terms. The sum of a finite series can be calculated, while the sum of an infinite series can only be calculated if the common ratio is less than 1.

Similar threads

I Question on an infinite summation series
    • Calculus
Replies
5
Views
431
B I don't recognize this limit of Riemann sum
    • Calculus
Replies
2
Views
1K
I Taylor Expansion Question about this Series
    • Calculus
Replies
2
Views
1K
I How does the ratio test fail and the root test succeed here?
    • Calculus
Replies
6
Views
840
MHB How do you find the sum of an infinite series?
    • Calculus
Replies
7
Views
1K
I On (real) entire functions and the identity theorem
    • Calculus
Replies
2
Views
891
I Understanding the nth-term test for Divergence
    • Calculus
Replies
15
Views
2K
I Does the sum of all series 1/n^m, m>1 converge?
    • Calculus
Replies
4
Views
2K
I Taming a Divergent Series -- But how does it work?
    • Calculus
Replies
14
Views
2K
A Is the following sum a part of any known generalized function?
    • Calculus
Replies
2
Views
992

Hot Threads

Recent Insights

Back
Top