If S1>S2, then limn →∞ Sn must be equal to (A) S1-S2 (B) S1+2 S2 (C) (S1+2 S2)/(3) (D) None of t

Name: Problem 127, Chapter 10: A Complete Resource Book in Mathematics for JEE Main
Uploaded: 2021-10-09T09:36:31-08:00
Duration: 1 min 42 s
Channel: Varsha Aggarwal
Description: If S1>S2, then limn →∞ Sn must be equal to (A) S1-S2 (B) S1+2 S2 (C) (S1+2 S2)/(3) (D) None of these

step 1 function we have to find limit of function 1 upon 1 minus n squared plus 2 upon 1 minus n square

Question

If $S_{1}>S_{2}$, then $\lim _{n \rightarrow \infty} S_{n}$ must be equal to (A) $S_{1}-S_{2}$ (B) $S_{1}+2 S_{2}$ (C) $\frac{S_{1}+2 S_{2}}{3}$ (D) None of these

   If $S_{1}>S_{2}$, then $\lim _{n \rightarrow \infty} S_{n}$ must be equal to
(A) $S_{1}-S_{2}$
(B) $S_{1}+2 S_{2}$
(C) $\frac{S_{1}+2 S_{2}}{3}$
(D) None of these

A Complete Resource Book in Mathematics for JEE Main

Dinesh Khattar 2019 Edition

Chapter 10, Problem 127

Instant Answer

Solved by Expert Varsha Aggarwal

10/09/2021

Step 1

We are given that $S_1 > S_2$, which means that the first term in the sequence is greater than the second term. We are also given that the sequence converges, which means that as $n$ goes to infinity, the terms in the sequence approach a specific value. Now, Show more…

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If $S_{1}>S_{2}$, then $\lim _{n \rightarrow \infty} S_{n}$ must be equal to (A) $S_{1}-S_{2}$ (B) $S_{1}+2 S_{2}$ (C) $\frac{S_{1}+2 S_{2}}{3}$ (D) None of these

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Key Concepts

Sequence Limits

This concept involves the idea of what happens to the terms of a sequence as the index becomes very large. It includes understanding the definition of the limit of a sequence and recognizing the conditions under which a sequence converges to a specific value.

Convergence of Recursively Defined Sequences

Many sequences are defined in terms of their previous terms using a recurrence relation. Understanding how such sequences converge involves identifying a fixed point—the value that remains unchanged when the recurrence relation is applied—and verifying that the recurrence leads the sequence toward that fixed point.

Monotonicity and Boundedness

A sequence is often easier to analyze if it is monotonic (either entirely non-increasing or non-decreasing) and bounded. The Monotone Convergence Theorem states that any sequence that is both monotonic and bounded must converge, making these properties critical when assessing the behavior of sequences.

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