show that the sequence defined by sn=(1/n+1)+(1/n+2)+(1/n+3)+…+(1/n+n) converges


To prove this we will show that this sequence is bounded as well as monotonic

First , we prove that it is bounded . for this, we have

\left | s_n \right |=s_n=\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+...+\frac{1}{n+n}< \frac{1}{n}+...+\frac{1}{n}\;\; up\;to\;n\;terms
=n.\frac{1}{n}=\Rightarrow \left | s_n \right |<1 \;\;for \;all\; n

thus , the sequence <sn> is bounded . Now , we have to prove that it is monotonic

for this we have

s_{n+1}-s_n=\left ( \frac{1}{n+2}+\frac{1}{n+3}+...+\frac{1}{2n+2} \right )-\left ( \frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n} \right )\\ \\ =\frac{1}{2n+1}-\frac{1}{2n+2}-\frac{1}{n+1}\\ \\ =\frac{1}{2n+1}-\frac{1}{2n+2}>0

Thus the sequence <s_n> is monotonically increasing . since <s_n> is bounded and monotonically increasing sequence hence its converges.

Related  science facts for students | Science Facts In Hindi Pdf | विज्ञान के अद्भुत तथ्य Science Facts In Hindi Images Science Facts that Will Blow Your Mind

Leave a Reply

Your email address will not be published. Required fields are marked *

Top 5 Most Expensive Domains Ever Sold 4 Must-Try ChatGPT Alternatives: Perplexity AI, BardAI, Pi, and More! Types of Trading Techniques in the Stock Market. ChatGPT app now available in India this AI chatbot can help you make your life more productive. What is wrong with following function code?