Limit at Infinity of a Sum

by Vikram Puri
(Noida, India)

Here's a problem about the limit at infinity of a sum. There is a very similar example at the limits at infinity main article. The limit is:



Here the strategy is the same as in the example on the main article. We realize that we can write this limit as:



And that is an arithmetic sum. The formula for an arithmetic progression is:



Pluggin that into our limit we get:



And now, factoring the denominator and simplifying:



And now we use the basic technique for finding limits at infinity. We divide everything by n:



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