Sum of Series: 1/(1.3.5)+1/(3.5.7)+1/(5.7.9)+...n Terms

[lakshya91]

sum up the series :
1/(1.3.5)+1/(3.5.7)+1/(5.7.9)+.......n terms.

 

[Dickfore]

The general term of the series is:

$$a_{n} = \frac{1}{(2 n - 1)(2 n + 1)(2 n + 3)}, \ n \ge 1$$

It can be represented in terms of partial fractions:

$$a_{n} = \frac{A}{2 n - 1} + \frac{B}{2 n + 1} + \frac{C}{2 n + 3}$$

Find $A$, $B$ and $C$!

After you do that, notice that the denominator of the second term is just that of the first term evaluated for $n + 1$ and the third is for $n + 2$. You can use some trick after that which is pretty common to simplify the n-th partial sum of the series

$$S_{n} = \sum_{k = 1}^{n} {a_{k}}, \ n \ge 1$$

 

[lakshya91]

The general term of the series is:

$$a_{n} = \frac{1}{(2 n - 1)(2 n + 1)(2 n + 3)}, \ n \ge 1$$

It can be represented in terms of partial fractions:

$$a_{n} = \frac{A}{2 n - 1} + \frac{B}{2 n + 1} + \frac{C}{2 n + 3}$$

Find $A$, $B$ and $C$!

After you do that, notice that the denominator of the second term is just that of the first term evaluated for $n + 1$ and the third is for $n + 2$. You can use some trick after that which is pretty common to simplify the n-th partial sum of the series

$$S_{n} = \sum_{k = 1}^{n} {a_{k}}, \ n \ge 1$$

hey, i tried this way but it does not worked out for me or may be i do not grasp your solution. Please give me a detailed solution

 

[uart]

hey, i tried this way but it does not worked out for me or may be i do not grasp your solution. Please give me a detailed solution

It's called a "telescoping" series. Why don't you start by showing us what numerical values you obtained for "A", "B" and "C" in the partial fractions expansion. Then we can tell you if you're on the right track or not.

 

[CellCoree]

I can approach this problem using mathematical principles and techniques. The given series is a finite geometric series, where each term is the product of the previous term and a constant ratio. In this case, the ratio is 1/3.

To find the sum of the series, we can use the formula for the sum of a finite geometric series:

S = a(1-r^n)/(1-r)

Where:
S = sum of the series
a = first term of the series
r = common ratio
n = number of terms

Using this formula, we can substitute the values from the given series:

S = (1/(1.3.5))(1-(1/3)^n)/(1-(1/3))

Simplifying this expression, we get:

S = (1/15)(1-(1/3)^n)/(2/3)

Next, we can substitute n with the number of terms in the series. For example, if there are 4 terms, n = 4.

Finally, we can solve for the sum S using basic arithmetic operations. The sum will depend on the number of terms in the series. As n approaches infinity, the sum will approach a finite value.

In conclusion, as a scientist, I can use mathematical principles to find the sum of the given finite geometric series. This approach can also be applied to other types of series in scientific research and data analysis.