a) Distinguishes  Between Convergent Sequences and Divergent Sequences

If {an} is convergent, when n increases to infinity, {an} is arbitrarily close to a constant number L. If {an} is divergent, when n increases to infinity, {an} won’t be arbitrarily close to a constant number L.

For example, {an}=1/n keeps decreasing when n is equal or bigger than 1, when n increases to infinity, {an}=1/n is arbitrarily close to 0.

So, we can say  {an}=1/n is convergent.

For example, {an}=(-1)^n, when n increases to infinity, {an}=(-1)^n is 1 or -1, which is not  arbitrarily close to L.

So, {an}=(-1)^n is divergent.

b) Distinguishes  Between Convergent Series and Divergent Series

A series is a infinite sum of sequences, if the infinite sum of sequences is arbitrarily close to a constant number L when n increases to infinite , we can say this series is convergent.

If the infinite sum of sequences is not arbitrarily close to a constant number L when n increases to infinite , we can say this series is divergent.

For example, the series of the infinite sum of {an)=(1/2)^n is convergent. Because for Σ(1/2)^n=(1/2+1/4+1/8+…1/2^n), 1/2Σ(1/2)^n=(1/4+1/8+1/16+…1/2^n+1/2^n+1). Σ(1/2)^n- 1/2Σ(1/2)^n= 1/2Σ(1/2)^n=1/2-1/2^n+1. Σ(1/2)^n=1-1/2^n. When n increases to infinite, Σ(1/2)^n converges to 1.

So, Σ(1/2)^n is convergent.

On opposite, the series of the infinite sum of {an)=n is divergent. Σn=1+2+3+4+5+,,,,,,+n, which is also increasing to infinity. the series of the infinite sum of {an)=n cannot be be arbitrarily close to a constant number L.

So, Σn is divergent.