Copyright

a) Give an explicit example of a series \sum_{k=1}^{\infty} a_{k} (that is, give an explicit...

Question:

a) Give an explicit example of a series {eq}\sum_{k=1}^{\infty} a_{k} {/eq} (that is, give an explicit formula for {eq}a_{k} {/eq}) such that {eq}\sum_{k=1}^{\infty} a_{k} = 5 {/eq} but {eq}\lim_{n \rightarrow \infty} s_{n} {/eq} diverges, or explain why such an example is impossible.

b) Give an explicit example of a series {eq}\sum_{k=1}^{\infty} a_{k} {/eq} (that is, give an explicit formula for {eq}a_{k} {/eq}) such that {eq}\sum_{k=1}^{\infty} s_{n} {/eq} diverges but {eq}\sum_{k=1}^{\infty} a_{k} {/eq} converges to {eq}3 {/eq}, or explain why such an example is impossible.

Divergence test:

The divergence test or also known as the nth term test states that if the limit of each term in the series approaches to zero as the term approaches infinity, then the sum of the series converges.

Answer and Explanation:

a)

The partial sum {eq}S_n {/eq} is a chunk of the sum of the original series evaluated from {eq}1 {/eq} to some number {eq}n{/eq}. Since the series converges to {eq}5 {/eq},

$$\sum_{k=1}^{\infty} a_{k} = 5 $$

it means that the limit of each term converges to zero

$$\lim _{k\to \infty }\left(a_k\right)=0. $$

Hence, for a partial sum {eq}S_n {/eq}, if we keep adding the term {eq}a_k {/eq} which gets closer and closer to zero as we approach infinity, then, it will be impossible for the partial sum to diverge as we approach infinity.

b)

Using the same reasoning from above, if the partial sum {eq}S_n {/eq} diverges, we expect that each term {eq}a_k {/eq} doesn't get closer and closer to zero which makes impossible for the series as a whole to converge by the nth term test.


Learn more about this topic:

Loading...
Infinite Series & Partial Sums: Explanation, Examples & Types

from GRE Math: Study Guide & Test Prep

Chapter 12 / Lesson 4
4.9K

Related to this Question

Explore our homework questions and answers library