Actually, itisn'tsmaller- it is "less than or equal to". That is, , for any x whatsoever, because for all x.

Those are both "geometric" sequences:I guess if I understood this it would make sense that as n tends to infinity, the fn. tends to 0.

I am also confused from reading wolfram articles how to evaluate a geometric series and how to tell whether one converges or not. Could someone maybe explain to me how to evalaute:

a) sum between 2&9 of 10^-n

&

b) sum between 0&ininfity of 3^(-n/2)

&

And, of course, .

Do you mean thec) determine whether convergent or not:

Between 1&infinity [(n-1)/n]sum? it's fairly easy to see that and thatsequenceconverges to 1 as n goes to infinity.

Do you know this theorem: If converges, then ?

If you you don't recognise that, realize that after n= 100000000000, say, you are still adding numbers close to 1.

many thanks