All Questions
Tagged with summation algebra-precalculus
90
questions with no upvoted or accepted answers
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Redefining a Variable in terms of itself vs creating a new one
Suppose I want to show
$$\sum_{n=1}^{\infty}ar^{n-1} = \sum_{n=0}^{\infty}ar^{n}$$
Is it acceptable to increment $n$ by redefining it in terms of itself, i.e., $n = n+1$ (written n+=1 in many ...
- 1,206
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Polynomials and power series. How to prove $\frac{1-x^{11}}{1-x}=1+x+x^2+x^3+\cdots+x^{10}$
How do I prove that $$\dfrac{1-x^{11}}{1-x}=1+x+x^2+x^3+\cdots+x^{10} $$
Attempt: By observing first let assume that $∣x∣<1$ then $\dfrac{1-x^{11}}{1-x}=\dfrac{1}{1-x}(1-x^{11})=(1+x+x^2+x^3+x^4+\...
- 2,034
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Solve for $x$: $\sum_{r=0}^5 {5\choose r} (-1)^rx^{5-r}3^r = 32$
Solve for $x$: $$\sum_{r=0}^5 {5\choose r} (-1)^rx^{5-r}3^r = 32$$
Looks like binomial theorem. So this would simplify to $(-x+3)^5=32$, and solving gives $x=2$. Is this correct?
- 6,915
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Proving an identity with geometric series.
I've been at this for MANY hours and I think it's time I sought help.
Question: Given $k = \frac{2 \pi}{Na}\left ( p-\frac{N}{2} \right )$, prove that $\sum_{k=1}^{N}e^{ika\left ( n-m \right )}=N\...
- 6,363
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Limit in sum and fraction
I'm staring at
$$ f(n) = \sum_{x=1}^n a(n)^{n-x}\\
a(n) = \frac{(1 - \frac{f}{n})y}{y(1 - \frac{f}{n} - \frac{\delta}{n}) + \frac{\delta}{n}}$$
where $f$, $y$, $\delta$, all $ \in (0, 1)$, and $n$ ...
- 1,089
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213
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Is there a summation formula for this equation (contains square roots, and functions within the square root)?
I am trying to solve a summation formula that is quite complex. However, to make the "answering" process for you guys easier I'll isolate the part I am having trouble with...
The equation is as ...
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1
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Some trouble with algebra using logarithms and summations
I'm having some embarrassing trouble with algebraic manipulation.
I have the function $$f(y) = y^Tx-\log\sum_{i=1}^ne^{x_i}$$
and for each $i = 1,2,\ldots,n$ $$y_i = {e^{x_1} \over \sum_{i=1}^ne^{...
user192348
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Quick simplification strategy for $\binom{3}{2}p^2(1-p) \le \sum_{k=3}^{5}\binom{5}{k}p^k(1-p)^{n-k}$
What is a quick simplification strategy to solve the following expression for $p$ by hand?
(or less preferably, by a TI83/86 calculator).
$$\binom{3}{2}p^2(1-p) \le \sum_{k=3}^{5}\binom{5}{k}p^k(1-p)...
- 3,082
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Simplify the math expression $M=(1-b)^{N}+(N-4)(1-b)^{N-1} b+ \sum^{N}_{i=1} 2^{2i-2} \Delta^2 (1-b)^{N-1}b-(1-2b)^2$
Can someone help me to further simplify the following expression? Here, $0<b<1$ and we can assume that $b$ is small. $\Delta$ is a constant. Thank you
$M=(1-b)^{N}+(N-4)(1-b)^{N-1} b+ \sum^{N}_{...
- 11
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417
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amortized analysis
a) define f(k) as the largest power of 2 that divides k.
For example, f(25) = 1, f(42) = 2, f(144) = 16.
What is ${1 \over k}\sum_1^k f(k)$?
b) define f(k) as the square of largest power of 2 that ...
- 147
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1
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Is a summation formula an object or just syntactic sugar?
This question is basically more aimed at an understanding of how certain things are formalized.
Is the addition for $\mathbb{Z}_{-k,n}$ and the $\sum_{i=-k}^n a_i$ considered as equivalent constructs? ...
- 1,609
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Query about summation in derivation of complex fourier series
I was trying to follow a derivation on the complex fourier series, but I am a bit confused at one particular step. In the following video https://www.youtube.com/watch?v=Ft5iyapkSqM, at 6:30, the ...
- 275
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What is the difference between the summations?
What is the difference between the summation $$\sum_{1 \leq i<j \leq n} f(i,j)$$ and $$\sum_{1\leq i} \sum_{<j \leq n} f(i,j)?$$
-1
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1
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Working through summations to show equation
Given
equation 1:
$$E = \sum_{k=1}^N \tau x_k g(\frac{n_k}{\tau}) + \sum_{k=1}^N n_kh(\frac{n_k}{\tau})$$
equation 2:
$$E = \frac{1}{2}\gamma X^2 + \epsilon \sum_{k=1}^N |n_k| +\frac{\eta*}{\tau}\...
- 123
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Is there a closed form for $\sum_{k=0}^n2^k\binom{2n+1}{2k}$, the sum of binomial coefficients times powers of two, for even indices?
I'm hoping for a nice and simple closed form for the sum
$$\sum_{k=0}^{n}2^k\binom{2n+1}{2k}.$$
Searching this site I found many nondescript titles but no duplicates, though I wouldn't be surprised if ...
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