**Infinite Series Explained in 5 Minutes**

An infinite series is just a sum of terms that never stops, for example..

s=1+(1/2)+(1/4)+(1/8)+(1/16)+..

This can be written more compactly as s=sum(1/2^n) for n=0 to infinity.

If the sum of a series is a finite number the series is said to converge, otherwise it is said to diverge. The example given above is a convergent series and in fact s=2.

It's not always easy to tell if a series converges. For example, the series s=1+(1/2)+(1/3)+(1/4)+(1/5)+.. looks like it converges but in fact it's a divergent series and s is not a finite number.

Infinite series are used a lot in mathematics because many functions can be written as infinite series, for example..

1/(1-x)=1+x+x^2+x^3+x^4..

e^x=1+x+(x^2/2!)+(x^3/3!)+(x^4/4!).. where n! is the factorial function n!=1*2*3*...*n

cos(x)=1-(x^2/2!)+(x^4/4!)-(x^6/6!)..

Series can be used to prove things about a function that might otherwise be difficult to prove. For example, if you differentiate the series for e^x term by term you get the same series! So this is a neat proof of the famous fact that e^x is its own derivative, d(e^x)/dx=e^x

Content written and posted by Ken Abbott abbottsystems@gmail.com