- #1
adamg
- 48
- 0
hey, i was just wondering if anyone could help find the sum of the infinite series defined by 1/[n(n+1)(n+2)]. I can split it into partial fractions but not sure from there. Thanks
Hurkyl said:What do you think of this decomposition?
[tex]
\frac{1}{k (k+1) (k+2)} = \frac{1}{2}
\left(
\frac{1}{k} - \frac{1}{k+1} - \frac{1}{k+1} + \frac{1}{k+2}
\right)
[/tex]
dextercioby said:According to Maple each of the three sums is:
[tex] \sum_{k=1}^{n}\frac{1}{2k}=\frac{1}{2}\psi(n+1)+\frac{1}{2}\gamma [/tex] (3)
[tex] \sum_{k=1}^{n} \frac{-1}{k+1}=-\psi(n+2)+1-\gamma [/tex] (4)
[tex] \sum_{k=2}^{n} \frac{1}{2(k+2)}=\frac{1}{2}\psi(n+3)-\frac{3}{4}+\frac{1}{2}\gamma [/tex](5)
Daniel.
The sum of an infinite series is the total value obtained by adding all the terms in the series together, from the first term to infinity.
To calculate the sum of an infinite series, you can use various methods such as the geometric series formula or the telescoping series method. These methods involve finding a pattern in the series and using mathematical formulas to determine the sum.
Yes, the sum of an infinite series can be negative. This can happen when the terms in the series alternate between positive and negative values, resulting in a negative overall sum.
The sum of an infinite series is important in mathematics as it helps us understand the behavior of a series and determine if it converges or diverges. It also has applications in various fields such as physics, engineering, and economics.
Yes, the sum of an infinite series can be infinite. This happens when the terms in the series increase in value without bound, resulting in a sum that also increases without bound. In this case, the series is said to diverge to infinity.