1
vote
1answer
58 views

Intuition behind a convergent subsequence of $\sin(n)$

$\let\eps\varepsilon$ I was looking through a Real Analysis exam paper one day and was stuck on a question; fortunately there is a solution provided which I will sketch below, but I have no intuition ...
3
votes
5answers
125 views

Why don't taylor series represent the entire function?

Say, I have a continuos function that is infinitely differentiate on the interval $I$. It can then be written as a taylor series. However, taylor series aren't always completely equal to the function ...
1
vote
1answer
79 views

What are the coefficients of these trigonometric sums?

I have two functions that I'm working on. The first is: $$ \begin{align} \cos x &= (\cos 1)^3 \cos(3-x) \\ &{}+ 3 (\cos 1)^2 (\sin 1) \sin(3-x) \\ &{}- 3 (\cos 1) (\sin 1)^2 \cos(3-x) \\ ...
15
votes
1answer
308 views

What is known about doubly exponential series?

I've been exploring functions that have a general form: $$\sum_{k=0}^\infty{ a^{b^k} } \tag{1}$$ In particular, I'm now checking this equality, which seems to hold: $$2 \sum_{k=0}^\infty{ \left( ...
2
votes
1answer
55 views

Finding Similar Sequences

Can we find two sequences: $$\{a (b^0), a (b^1), a (b^2), a (b^3), \dots, a(b^n)\} \bmod p_1$$ $$\{c (d^0), c (d^1), c (d^2), c (d^3), \dots, c(d^n)\} \bmod p_2$$ that differ by only one number? ...
1
vote
1answer
65 views

Expand $\left(1-\frac2x\right)^{\frac12}$ to find a value of $\sqrt{99}$ and $\sqrt{101}$

Given that $|x|\gt2$ find the first four terms in the series expansion of $\left(1-\frac2x\right)^{\frac12}$ in descending powers of $x$. By taking $x$ = 200 use the series to find a value of ...
9
votes
2answers
455 views

Intuition for the Frobenius method

I'm teaching a differential equations class now and I am hoping to give a reason for the Frobenius series method beyond simply "we guess these solutions". Now, for the Euler equation $$t^n x^{(n)}(t) ...
9
votes
5answers
430 views

how do I derive $1 + 4 + 9 + \cdots + n^2 = \frac{n (n + 1) (2n + 1)} 6$ [duplicate]

Possible Duplicate: Proof that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$? I am introducing my daughter to calculus/integration by approximating the area under y = f(x*x) by ...
2
votes
2answers
478 views

Get sum to a closed form ideas on how to start

I have this sum $ 1^2+ 2^2 + 3^2 + \ldots + x^2$ I started getting some sums to their closed forms. But I see that sometimes I start off on the bad track. So I'd like if it's possible some tips on ...
1
vote
0answers
61 views

A gratifying re-encounter with a piece of math that was out of my mind

A series of real numbers is said to be conditionally convergent if it is convergent but not absolutely convergent. By rearranging the terms of a conditionally convergent series we can make the ...
10
votes
3answers
595 views

What could be an intuitive explanation for $ \sum\limits_{k=1}^{\infty}\frac{1}{k2^k} = \log 2 $?

What could be an intuitive explanation for $\displaystyle \sum_{k=1}^{\infty}{\frac1{k\,2^k}} = \log 2$ ? $\displaystyle \sum_{k=1}^{\infty}{\frac{1}{2^k}} = 1$ has a simple intuitive explanation ...
3
votes
1answer
798 views

Why is the ratio test for L=1 inconclusive?

One of the often used tests for convergence (L<1) and divergence (L>1) of an infinite series is the ratio test. The idea behind it, why it works is the geometric series which dominates (or not) ...
8
votes
0answers
472 views

Combinatorial reasoning for the identity $\left ( \sum_{i=1}^n i \right )^2 = \left ( \sum_{i=1}^n i^3 \right ) $ [duplicate]

Possible Duplicate: Intuitive explanation for the identity $\sum\limits_{k=1}^n {k^3} = \left(\sum\limits_{k=1}^n k\right)^2$ There is the interesting identity: $$\left ( \sum_{i=1}^n i ...
-2
votes
1answer
115 views

Concerning $a_{f(n)} = \displaystyle\sum_{i=0}^m{c_i a_{g_i(n)}}$

Suppose $f(n)$ is $2n$. Suppose $g_i(n)$ is $n+i$. In other words, we have a recurrence given by $a_{2n} = c_0 a_n + c_1 a_{n+1} + \dots c_m a_{n+m}$ (1) What is known about this? (2) What type of ...
2
votes
0answers
436 views

Numerical Methods: Approximating an Integral with Exponentials [closed]

I'd like to know about the best numerical methods for approximating an integral. Unfortunately, I want to know about a fairly general case, so I cannot give a lot of information. Essentially, I have ...
5
votes
3answers
721 views

Convergence of the series $\sum \limits_{n=2}^{\infty} \frac{1}{n\log^s n}$

We all know that $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s}$ converges for $s>1$ and diverges for $s \leq 1$ (Assume $s \in \mathbb{R}$). I was curious to see till what extent I can push the ...
0
votes
1answer
71 views

Constructing finite versions of arbitrary series

I've been wondering if/when it's possible to "truncate" a series. Example 1 For example, the closed form for the series of naturals is: $\frac{1}{(x-1)^2}$ = $1 + 2z + 3x^2 + \cdots$ The ...
0
votes
1answer
127 views

Is there an easier formulation for the Hadamard product of certain pair of series?

I'm interested in finding a recursion or simple representation for a "Hadamard product" of two power series. The Hadamard Product The Hadamard product is defined on generating functions $f(x)$ ...
7
votes
1answer
152 views

Is there a relationship between e and the sum of n-simplexes volumes?

When I look at the Taylor series for e^x and the volume formula for oriented simplexes, it makes e^x look like it is, at least almost, the sum of simplexes volumes from n to infinity. Does anyone know ...