Summation of a function

mtwisd

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Hello,

I am familiar with the Summation Formulas. I am having difficulty applying them to Σ (1/(k^2 +1)), k, 1, 5).

Could someone help me answer this question? I am assuming that I have to manupulate the function to fit the Summation Formulas format but, I do not know how.

Best,
MTWISD
 
Hello,

I am familiar with the Summation Formulas. I am having difficulty applying them to Σ (1/(k^2 +1)), k, 1, 5).

Could someone help me answer this question? I am assuming that I have to manupulate the function to fit the Summation Formulas format but, I do not know how.

Best,
MTWISD
What does 1/(k^2 +1) equal when K=1? What does 1/(k^2 +1) equal when K=2?...What does 1/(k^2 +1) equal when K=5? What do you do with these results?
 
I am familiar with the Summation Formulas. I am having difficulty applying them to Σ (1/(k^2 +1)), k, 1, 5).
By "the Summation Formulas", I will guess that you mean that your book has provided you with some basic and common summation results, such as:

. . . . .\(\displaystyle \displaystyle{ \sum_{i\, =\, 1}^n\, c\, =\, cn }\)

. . . . .\(\displaystyle \displaystyle{ \sum_{i\, =\, 1}^n\, i\, =\, }\) \(\displaystyle \dfrac{n\,(n\, +\, 1)}{2}\)

. . . . .\(\displaystyle \displaystyle{ \sum_{i\, =\, 1}^n\, i^2\, =\, }\) \(\displaystyle \dfrac{n\, (n\, +\, 1)\,(2n\, +\, 1)}{6}\)

. . . . .\(\displaystyle \displaystyle{ \sum_{i\, =\, 1}^n\, i^3\, =\, }\) \(\displaystyle \dfrac{n^2\, (n\, +\, 1)^2}{4}\)

The left-hand side of each equality above is in "summation" notation; the right-hand side of each equality above is in the "closed" form. In other words, you'd have to plug in each value of "i", evaluate, and simplify to find each term's value, and then sum to find the summation value, for the left-hand side. For the right-hand side, you'd only have to plug in the final value of the counter.

By "Σ (1/(k^2 +1)), k, 1, 5)", I will guess that you are needing to find the sum, from k = 1 to k = 5, of the fractional expression, so the summation is as follows:

. . . . .\(\displaystyle \displaystyle{ \sum_{k\, =\, 1}^5\, }\) \(\displaystyle \,\dfrac{1}{k^2\, +\, 1}\)

By "applying [the Summation Formulas]", I will guess that you are being expected to express the given summation in terms of the closed-form expressions you have been provided. However, none of the formulas I've displayed above would apply to what you've posted. Please reply with all of the summation formulas they've given you, so we can see which one(s) might apply apply.

When you reply, please include any corrections of my assumptions (stated above), along with a clear listing of your efforts so far. Thank you! ;)
 
Frankly, since you are only adding 5 numbers, I would think it foolish to use general formulas.

\(\displaystyle \displaystyle \sum_{k=2}^5 \frac{1}{k^2+ 1}= \frac{1}{1^2+ 2}+ \frac{1}{2^2+ 1}+ \frac{1}{3^2+ 1}+ \frac{1}{4^2+ 1}+ \frac{1}{5^2+ 1}\)
\(\displaystyle = \frac{1}{3}+ \frac{1}{5}+ \frac{1}{10}+ \frac{1}{17}+ \frac{1}{26}\)

The only difficulty is that the "least common denominator" is 2(3)(5)(13)(17)= 6630.
 
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