Hi Show that the sequence given by an = 1/(n+1) + (1/(n+2)) + (1/(n+3)) +...... (1/(n+n')) converges.
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Originally Posted by champrock an = 1/(n+1) + (1/(n+2)) + (1/(n+3)) +...... (1/(n+n')) converges. Have you noticed that $\displaystyle a_n = \sum\limits_{k = 1}^n {\frac{{\frac{1}{n}}}{{1 + \frac{k}{n}}}} $? That is an approximating sum for $\displaystyle \int_0^1 {\frac{{dx}}{{1 + x}}} $.
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 ?
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