limn →-[(n)/(n^2)+(n)/(n^2+1^2)+(n)/(n^2+2^2)+…+(n)/(n^2+(n-1)^2)] is equal to (a)-(π)/(4) (b) 0

Name: Problem 4, Chapter 6: Higher Engineering Mathematics
Uploaded: 2021-10-01T02:21:48-08:00
Duration: 1 min 23 s
Channel: Aman Gupta
Description: limn →-[(n)/(n^2)+(n)/(n^2+1^2)+(n)/(n^2+2^2)+…+(n)/(n^2+(n-1)^2)] is equal to (a)-(π)/(4) (b) 0 (c) (π)/(4) (d) (π)/(3)

step 1 in this problem we have to find limit of n tends to infinity 1 over n 1 plus n square upon n squ

Question

$\lim _{n \rightarrow-}\left[\frac{n}{n^{2}}+\frac{n}{n^{2}+1^{2}}+\frac{n}{n^{2}+2^{2}}+\ldots+\frac{n}{n^{2}+(n-1)^{2}}\right]$ is equal to $(a)-\frac{\pi}{4}$ (b) 0 (c) $\frac{\pi}{4}$ (d) $\frac{\pi}{3}$

   $\lim _{n \rightarrow-}\left[\frac{n}{n^{2}}+\frac{n}{n^{2}+1^{2}}+\frac{n}{n^{2}+2^{2}}+\ldots+\frac{n}{n^{2}+(n-1)^{2}}\right]$ is equal to
$(a)-\frac{\pi}{4}$
(b) 0
(c) $\frac{\pi}{4}$
(d) $\frac{\pi}{3}$

Higher Engineering Mathematics

Grewal B.S. 42nd Edition

Chapter 6, Problem 4

Instant Answer

Solved by Expert Aman Gupta

10/01/2021

Step 1

Step 1: We start with the limit expression given in the problem: \[ \lim_{n \to \infty} \left[\frac{n}{n^2} + \frac{n}{n^2 + 1^2} + \frac{n}{n^2 + 2^2} + \ldots + \frac{n}{n^2 + (n-1)^2}\right] \] Show more…

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$\lim _{n \rightarrow-}\left[\frac{n}{n^{2}}+\frac{n}{n^{2}+1^{2}}+\frac{n}{n^{2}+2^{2}}+\ldots+\frac{n}{n^{2}+(n-1)^{2}}\right]$ is equal to $(a)-\frac{\pi}{4}$ (b) 0 (c) $\frac{\pi}{4}$ (d) $\frac{\pi}{3}$

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Key Concepts

Limit of a Sequence

This concept involves analyzing the behavior of a sequence as its index approaches infinity. Limits help us understand convergence and determine the ultimate value that a sequence or function approaches, which is fundamental to calculus and analysis.

Riemann Sum

A Riemann sum approximates the value of a definite integral by dividing the region under a curve into smaller segments or subintervals, summing the product of the function value at specific points and the width of each subinterval. This method transitions smoothly into the exact computation of an integral as the number of subintervals increases.

Definite Integral

A definite integral represents the exact accumulation of quantities, such as area under a curve, between two bounds. Its value is defined as the limit of Riemann sums as the partition of the interval becomes infinitely fine, making it a central tool in calculus for solving problems related to accumulation and area.

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