The mathematical expression

means

More generally,

means

.

But what if you don’t understand what “+ … +” means? Sure, most people understand what that means intuitively, but how could you *define* “…”?

From a logical point of view, you could instead make a recursive definition

and

.

Most of the time, when I want to prove a summation, I prove it by induction anyway, so this definition is actually helpful. Suppose we want to prove

For ,

by definition.

Supposing the summation to work for , for the case we have

where the first equation comes from the recursive definition of the summation, the second by hypothesis, and the third by algebra.

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Tags: induction, recursion, summation notation

This entry was posted on May 28, 2009 at 12:23 pm and is filed under math. You can follow any responses to this entry through the RSS 2.0 feed.
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