Problem 69
Question
State the definitions of convergent and divergent series.
Step-by-Step Solution
Verified Answer
A convergent series is a series whose sequence of partial sums tends to a finite limit. Conversely, a divergent series is a series whose sequence of partial sums does not tend to a finite limit.
1Step 1: Define Convergent Series
A series \( \sum a_n \) is said to be convergent if the sequence of its partial sums \( S_n = a_1+a_2+a_3+...+a_n \) tends to a finite limit as \( n \rightarrow \infty \). That is, if there exists a finite number 'l', such that for every real number \( \epsilon >0 \), there exists a positive integer \( N \) such that for all \( n \geq N\), \( |S_n-l|< \epsilon \), then the series \( \sum a_n \) is said to be convergent series.
2Step 2: Define Divergent Series
If a series \( \sum a_n \) is not convergent, i.e., the sequence of its partial sums \( S_n = a_1+a_2+a_3+...+a_n \) does not tend to a finite limit as \( n \rightarrow \infty \), then the series is said to be divergent.
Other exercises in this chapter
Problem 68
(a) Find the power series centered at 0 for the function \(f(x)=\frac{\ln \left(x^{2}+1\right)}{x^{2}}\). (b) Use a graphing utility to graph \(f\) and the eigh
View solution Problem 68
Use the Root Test to determine the convergence or divergence of the series. $$ \sum_{n=0}^{\infty} e^{-n} $$
View solution Problem 69
Determine whether the sequence with th given \(n\) th term is monotonic. Discuss the boundedness of th sequence. Use a graphing utility to confirm your results.
View solution Problem 69
Let \(f(x)=\sum_{n=0}^{\infty} c_{n} x^{n},\) where \(c_{n+3}=c_{n}\) for \(n \geq 0\). (a) Find the interval of convergence of the series. (b) Find an explicit
View solution