Find a non-summation expression for the value of the sum .

To find the sum , we first note that , so that our sum . Next, we look back at this post, where it is demonstrated that

Using a similar method, we see that

And thus, for ,

which, using the quotient rule, gives us

However, we can express this more compactly using trigonometric identities. With the product-to-sum formulas, we see:

and

Substituting these into the numerator, and expanding, we see

which is much more compact.

Lastly, one could also use the half-angle identity to give equivalent form

Now, when *x* is an integer multiple of 2*π*. For these values, since *n* is an integer, , and so our expression has the indeterminate form 0/0. Thus, we can apply l’Hôpital’s rule (twice) for the limit as *x* approaches zero (and, thus due to the period, any of the other singularities):

so the singularities are removable, with a limit which matches the value expected from the sum:

for *x*=0, .

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Tags: Calculus, L'Hôpital's Rule, Math, Monday Math, Telescoping Series, Trigonometry

This entry was posted on August 11, 2014 at 12:33 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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August 11, 2014 at 7:45 am |

Reblogged this on Math Snippets and commented:

Nice Blog! 🙂 Check out new posts – very interesting Geometry and algebraic insights – every Monday! 🙂

August 11, 2014 at 7:57 am |

Hello! 🙂 I am just starting out with blogs (in particular Math ones)… Are there any tips you could share for posting such awesome posts??? 🙂