Question

Converges or diverges, find the sum. 3 -3/4 +3/16 -3/64.

Answer 1

We know the terms in this geometric sequence are 3, -3/4, 3/16 and so on.

now.. what's the "r" or common ratio?  well, in a geometric sequence, to get the next term's value you multiply by "r", therefore, if you divide any of those values, by the value before it, the quotient will be, yes, you guessed it, "r".

namely say a, a*b.... now (a*b)/a is just b, the quotient is the value you used as multiplier.

$\bf -\cfrac{3}{4}\div 3\implies -\cfrac{3}{4}\div \cfrac{3}{1}\implies -\cfrac{3}{4}\cdot \cfrac{1}{3}\implies -\cfrac{1}{4}$

any other two pair of values will spit out the same "r".

now, the Sum of a geometric sequence reaches a limit or point, that is converges, only if the value of "r" is a fraction and more than -1 or less than 1, negative or positive.

or in short, if | r | < 1.

now, in this case, -1/4 is indeed more than -1 and less than 0, therefore, the Sum of the sequence is convergent, it meets at a limit.