Converging And Diverging Sequences Introduction

Lets take the Sequence {1, -1/2, 1/3, -1/4, 1/5…..}

Notice how the sequence is taking 1/ and dividing it by n(the number of the term) and then isolating between positive and negative. So first of all lets graph this to get an visual idea.

So after looking at this graph we can see each times the points are getting closer to 0.

But first lets write an equation for this….

so we can see were dividing one by the number of the term so we get 1/n but the numbers isolate from positive to negative. so we can multiply by -1 and raise power it to the n + 1 term so the numbers isolate from positive to negative with the first term being positive.

So our finial equation is 1/n * -1^n+1

Now lets try to find the limit of this sequence….

well we can say the limit of an = the limit of 1/n * -1^n+1 as n tends to infinity

So as the n gets larger the numbers will get smaller making it so that teh sequence approaches 0.

So we can say an converges to 0.

This is essentially what it means for a sequence to converge to 0.

As the n of the sequence approaches infinity the number the sequence is going to is the number it converges to.

If an didn’t approach sum number we would say an diverges.

NOTE:

• Whenever I wrote an I meant a sub n.