To solve the problem of determining the sum \(\frac{3}{1^2 \times 2^2} + \frac{5}{2^2 \times 3^2} + \frac{7}{3^2 \times 4^2} + \ldots\), we first need to look for a pattern or a general term in the series.
Let's examine the terms clearly:
- The numerator seems to increase by 2 each time: \(3, 5, 7, \ldots\)
- The denominator is a product of consecutive squares, which can be represented as \(n^2 \times (n+1)^2\).
Given this analysis, the general term of the series can be expressed as:
\(\frac{2n + 1}{n^2 \times (n+1)^2}\)
Now, let's reconsider the series using partial fraction decomposition, which is often a useful strategy for managing such sequences:
The expression \(\frac{2n + 1}{n^2(n+1)^2}\) can be written as a difference of two terms:
\( \frac{1}{n^2} - \frac{1}{(n+1)^2} \)
This decomposition suggests a telescopic series, where most terms cancel out except the first and last ones as \(n\) approaches infinity. Let's write out a few terms:
- For n=1:
\(\frac{1}{1^2} - \frac{1}{2^2} = 1 - \frac{1}{4} = \frac{3}{4}\)
- For n=2:
\(\frac{1}{2^2} - \frac{1}{3^2} = \frac{1}{4} - \frac{1}{9} = \frac{5}{36}\)
Adding these and continuing to higher \(n\), we observe that the series collapses, leaving:
\( 1 - \lim_{n \to \infty} \frac{1}{(n+1)^2} = 1 \)
Thus, the value of the infinite sum is indeed 1.
Therefore, the correct answer is 1.
Similar Questions
What is \(\frac{1}{\sqrt{10}+\sqrt{9}} + \frac{1}{\sqrt{11}+\sqrt{10}} + \frac{1}{\sqrt{12}+\sqrt{11}} + \dots + \frac{1}{\sqrt{196}+\sqrt{195}}\) equal to ?
- There are \(n\) concentric squares. The area of the innermost square is 1 unit and the distance between corresponding corners of any two consecutive squares is 1 unit. Consider the following statements :
I. The diagonal of the \(n\)th square is \(2n+\sqrt{2}-2\)
II. The area included between \(n\)th square and \((n-1)\)th square is independent of \(n\)
Which of the statements given above is/are correct?
Important Questions from Sequence & Series
Which of the following options should come in place of the question mark (?) in the given series based on the English alphabetical order?
SHK, WLO, APS, ETW,?
Which of the following options should come in place of the question mark (?) in the given series based on the English alphabetical order?
EGI, KMO, QSU, WYA,?
Which of the following options should come in place of the question mark (?) in the given series?
14 15 18 23 30?
Two sets of numbers are given below. In each set of numbers, certain mathematical operation(s) on the first number result(s) in the second number. Similarly, certain mathematical operation(s) on the second number result(s) in the third number and so on. Which of the given options follows the same set of operations as in the given sets? (NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into their constituent digits. E.g. 13 – Operations on 13 such as adding/subtracting/multiplying to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed.)
What should come in place of the question mark (?) in the given series based on the English alphabetical order?
TAW VCY XEA ZGC ?
