Consider the infinite geometric in this image the lower limit of the summation notation is "n=1"
a. write the first four terms
b. does the series diverge or converge
c. if the series has a sum, find the sum

really need on this! i want an explanation !

A) The first four terms in the series are .

b) The series is geometric, and any geometric series converges if the ratio between common terms, , satisfies , which is the case here as .

c) The sum of the first terms of a geometric sequence is

Multiply both sides by :

Now subtract and you can solve for :

For an infinite geometric series, you're considering what happens as . Call this sum , so that

This limit will only exist when .

Let's write this sum using sigma notation.

Your series starts at , so to get this to be the same we can just shift the index.

Now, matching this up with your given series, it's clear that and , which is indeed between -1 and 1, so the series must converge. Indeed,