Expert: Paul Klarreich - 3/7/2010

Question
For what values of q>0 is the following series convergent, and find its sum:
1 + q + (q^2)/2 + (q^3)/3 + ... + (q^n)/n

I'm fairly certain that it is convergent for 0 < q < 1, but I don't know how to find the sum and have not been able to find anything helpful online or in my text/lecture notes.

This is for an introductory analysis course, and we are currently studying infinite series and sequences. Thank you, in advance.

Answer
Questioner: Ray
Country: United States
Category: Advanced Math
Private: No
Subject: series convergence
Question: For what values of q>0 is the following series convergent, and find its sum:
1 + q + (q^2)/2 + (q^3)/3 + ... + (q^n)/n

I'm fairly certain that it is convergent for 0 < q < 1, but I don't know how to find the sum and have not been able to find anything helpful online or in my text/lecture notes.

This is for an introductory analysis course, and we are currently studying infinite series and sequences. Thank you, in advance.
..............................

1 + q + (q^2)/2 + (q^3)/3 + ... + (q^n)/n

You can use the ratio test:

      |a[n+1]|
r[n] = ---------
       |a[n]|

If lim(n->inf) r[n] < 1, the series converges.
If lim(n->inf) r[n] > 1, the series diverges.
If lim(n->inf) r[n] = 1, we don't know.

In this case:

       q^(n+1)/(n+1)
r[n] = --------------
          q^n/n

        qn
r[n] =  -----
        n+1

And lim of that = q.

So you need q < 1.  

Of course, if q = 1, you have the harmonic series.

I don't know any calculation for the actual sum.