calculus - How to find the sum of an alternating series?
The following is a Taylor Series evaluated a particular value of x, find the sum of the series. 1.
Is there a general formula for the sum of a quadratic sequence?
Mar 13, 2014 · Quadratics have a sum which is a cubic equation, so take any four points and do Lagrange interpolation. When the kth difference of a sequence is constant, one can write down the general term using Newton's formula. f(n) = f(0) + Δf (0) 1! n + Δ2f (0) 2! n(n − 1) + ⋯, where Δf are the successive differences. For quadratic sequences, one ...
Finding the sum of a series - Mathematics Stack Exchange
Add a comment. 2. Hint for the first series: Expand it as a sum of geometric series. This is the most straightforward way to solve this, though there are others. Hint for the second series: Write the first few terms and see what you get. Edit: Here's how the first hint could be used: ∑n=1∞ nxn = ∑m≥1∑n≥mxn =∑m≥1 xm 1 − x = x ...
How can I define $e^x$ as the value of infinite series?
One definition of the exponential function is the limit. limn→∞(1 + x n)n =ex. lim n → ∞ (1 + x n) n = e x. limn→∞Pn(x) =∑n=0∞ xn n!. lim n → ∞ P n (x) = ∑ n = 0 ∞ x n n!. Pn(x) = ∑k=0n (n k)xk nk. P n (x) = ∑ k = 0 n (n k) x k n k. Now.
trigonometry - How can we sum up $\sin$ and $\cos$ series when …
$\begingroup$ Hint: reverse the series and sum it up term by term with the original series. So $\cos(a)+\cos(a+(n-1)\cdot d)$, etc... And use the Simpson formula for sums of cosines (and sines for the other identity). $\endgroup$ –
The idea behind the sum of powers of 2 - Mathematics Stack …
Oct 29, 2016 · There's a lot of visualisations of this out there. Write the numbers in base 2: The powers of 2 starting from 1 = 20 will be in binary, 1 + 10 + 100 + 1000 will always be a number that will be a n with all binary digits 1. This is the largest number having that many digits. SO it is of the form 2n + 1 − 1.
Sum of a power series $n x^n$ - Mathematics Stack Exchange
Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Sum of a series of a number raised to incrementing powers
Dec 28, 2014 · late to the party but i think it's useful to have a way of getting to the general formula. this is a geometric serie which means it's the sum of a geometric sequence (a fancy word for a sequence where each successive term is the previous term times a fixed number). we can find a general formula for geometric series following the logic below
analysis - Sum of the alternating harmonic series …
The odd partial sums and the even partial sums must approach the same limit, since the difference between each odd partial sum and the even partial sum before it approaches 0: $\lim\limits_{n \to \infty}\frac{1}{n} = 0$. Similarly, the difference between each even partial sum and the odd partial sum before it approaches 0. ∴ the series converges.
calculus - relation between integral and summation - Mathematics …
Aug 21, 2012 · 4. There is a certain relation between integral and summation given by ∑ i ∈ If(i) = ∫ If(x)#(dx) where # is the counting measure, and the function f is e.g. non-negative. Such connection is also described in article mentioned by FrenzY DT. However, in the previous version of yours the integral and summation are used for, say, orthogonal ...