- #1
physsure
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How would you, personally, do this summation the quickest way?
You can use the idea by Gauss ( when he was 8 years or so) . Pair up:physsure said:How would you, personally, do this summation the quickest way?
physsure said:How would you, personally, do this summation the quickest way?
@fresh_42 I believe your 3rd term needs a minus in front of it. ## \\ ## Note: ## \sum\limits_{k=1}^{n} k^2=\frac{(2n+1)(n+1)(n)}{6} ## is the most difficult term to evaluate in the above. The rest is relatively straightforward.fresh_42 said:$$50 \,\pi + \dfrac{25\cdot 67 \cdot 101}{\pi} + \dfrac{1}{2\pi}\sum_{k=1}^{100} (\pi-k)^2$$
Yep, thanks. Lost while turning pages in my scribble book.Charles Link said:@fresh_42 I believe your 3rd term needs a minus in front of it. ## \\ ## Note: ## \sum\limits_{k=1}^{n} k^2=\frac{(2n+1)(n+1)(n)}{6} ## is the most difficult term to evaluate in the above. The rest is relatively straightforward.
How about:Charles Link said:The rest is relatively straightforward.
I do think @WWGD probably had the best answer for the OP in post 3.fresh_42 said:How about:
$$
\dfrac{\sqrt{5}^9\cdot 101}{2\,\pi^2} \cdot \sum_{n=1}^{\infty} \left( \dfrac{1}{n} \right)^2\cdot \dfrac{1}{n+1}\cdot \dfrac{F_n\cdot L_n}{C_n}
$$
with the Fibonacci sequence ##F_n##, the Lucas sequence ##L_n##, and the Catalan sequence ##C_n##.
A summation, also known as a series, is the result of adding together a sequence of numbers. It is represented by the symbol "∑" and has a starting point, an end point, and a pattern for adding each number in between.
The quickest way to calculate a summation is by using a formula. This formula is known as the summation formula and it allows you to find the sum of a series without having to add each number individually.
The summation formula is: ∑(n = a to b) of f(n) = f(a) + f(a+1) + ... + f(b). This means that you take the starting number (a) and plug it into the function (f), then add the result to the function of the next number, and continue until you reach the end number (b).
If the numbers in the summation are not in a sequence, you will need to find a way to represent the series as a sequence. This could involve finding a pattern or using other mathematical techniques to simplify the series into a sequence.
Yes, there are other methods for quickly calculating a summation. One method is using a calculator or a computer program. Another method is using mathematical techniques such as telescoping or rearranging the series to make it easier to calculate. It is important to choose the method that is most efficient for the specific summation you are trying to solve.