Limits of a Sequence with the Sum of Squares or the Sum of Cubes
Krystian Karczyński
Founder and General Manager of eTrapez.
Graduate of Mathematics at Poznan University of Technology. Mathematics tutor with many years of experience. Creator of the first eTrapez Courses, which have gained immense popularity among students nationwide.
He lives in Szczecin, Poland. He enjoys walks in the woods, beaches and kayaking.
In the limits of sequences, it sometimes looks like this:
and sometimes even like this:
So what then?
The answer is simple:
formulas for the sum of squares and the sum of cubes of consecutive natural numbers.
They go like this:
Formulas – just like any formulas – are to be memorized. If you’ve had similar examples and really need them.
Knowing these formulas, calculating our limits becomes trivial:
Next limit:
Inductive proofs for formulas
The validity of the formulas can be quite easily proven inductively (at least a few years ago this was a complete standard in high school). I will do this for the formula:
1. 
Step 1 of induction
We check the validity of the formula for n=1:
It matches
Step 2 of induction
We assume assumption, that for a certain natural n:
Step 3 of induction
We prove the thesis (using the accepted assumption), that for n+1 the formula also holds, i.e.:
On the left side instead of 
we substitute the formula from the assumption, on the right we just organize:
And then instead of forcing it, we work a little more subtly:
So the thesis is proven. The formula is proven inductively.
I invite you to inductively prove the second formula, for the sum of cubes:
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