# Sequences question

• April 16th 2009, 12:09 PM
champrock
Sequences question
Hi

Show that the sequence given by

an = 1/(n+1) + (1/(n+2)) + (1/(n+3)) +...... (1/(n+n')) converges.
• April 16th 2009, 01:22 PM
Plato
Quote:

Originally Posted by champrock
an = 1/(n+1) + (1/(n+2)) + (1/(n+3)) +...... (1/(n+n')) converges.

Have you noticed that $a_n = \sum\limits_{k = 1}^n {\frac{{\frac{1}{n}}}{{1 + \frac{k}{n}}}}$?

That is an approximating sum for $\int_0^1 {\frac{{dx}}{{1 + x}}}$.
• April 16th 2009, 11:13 PM
champrock
i am slightly confused. cant this sequence be considered as an extension of the 1 + 1/1 + 1/2 + 1/3 sequence?

So the sequence goes like 1 + 1/1 + 1/2 + 1/3 ..... 1/(n+1) + (1/(n+2)) + (1/(n+3)) +...... (1/(n+n')) ...... 1/(n+n'+1) .... and so on ?