Question
Let $b_{n}=a_{n+1} .$ Use the limit definition to prove that if $\left\{a_{n}\right\}$ converges, then $\left\{b_{n}\right\}$ also converges and $\lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} b_{n}$
Step 1
This means that for any $\epsilon > 0$, there exists a natural number $N$ such that for all $n \geq N$, $|a_{n} - L| < \epsilon$, where $L$ is the limit of the sequence $\{a_{n}\}$. Show more…
Show all steps
Your feedback will help us improve your experience
Linh Vu and 58 other Calculus 1 / AB educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Let $\left\{b_{n}\right\}$ be a sequence and let $a_{n}=b_{n}-b_{n-1} .$ Show that $\sum_{n=1}^{\infty} a_{n}$ converges if and only if $\lim _{n \rightarrow \infty} b_{n}$ exists.
INFINITE SERIES
Summing an Infinite Series
Suppose that $b_{n}>0$ for all $n$ and $\sum b_{n}$ converges. Show that if $\lim a_{n} / b_{n}=0$ then $\sum a_{n}$ converges.
Sequences and Series
Tests for Convergence
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD