7 votes

Convergence of $\sum \frac{b_n}{n}$ Where $b_n = 1, -1, -1, 1, 1, 1, -1, -1,-1,-1,1,1,1,1,1,....$

This is equivalent to whether: $$\sum_{n=0}^\infty (-1)^n c_n$$ converges, where $$c_n=\sum_{k=T_{n}+1}^{T_{n+1}}\frac1{k}=\sum_{j=1}^{n+1}\frac{1}{T_n+j}$$ Where $T_n=1+2+\cdots +n=\frac{n(n+1)}{2}.$ ...
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6 votes
Accepted

Computing $\int_{0}^{1} \frac{x^{t}-1}{\ln x} d x$ without Feynman, where $Re(t)>-1$?

Sure, we can use the dual to Feynman - turning the integral into a double integral: $$\int_0^1\frac{x^t-1}{\log t}\:dx = \int_0^1\int_0^tx^y\:dy\:dx = \int_0^t \frac{1}{y+1}\:dy = \log(y+1)$$ by ...
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  • 25.8k
3 votes

Solve the ordinary differential equation $\frac{d^2}{dx^2}F(x)=\frac{1}{F(x)^2}$

If we write $y = F(x)$ and regard $x$ as a function of $y$, the differential equation becomes $$x'' = -\frac{(x')^3}{y^2},$$ where $'$ denotes $\frac{d}{dy}$. This equation is separable and first-...
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2 votes
Accepted

Root in $(1,2]$ of Equation $x^n-x-n=0$

First, I shall prove a stronger claim. Claim: $u_n \geq 1 + \frac{2}{n}$ for $n \geq 2$. Proof: Applying IVT on $\left(1 + \frac{2}{n}, 2\right)$, we simply have to evaluate $f_n\left(1 + \frac{2}{n}\...
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  • 2,096
2 votes

Spivak's Calculus, Ch. 13, Problem 39: Interpretation of Proof of Cauchy-Schwarz Inequality

You're absolutely right, and this is why on the space of Riemann-integrable functions, $f\mapsto \sqrt{\int_a^bf^2}$ is not a norm; it's only a semi-norm (equivalently, $(f,g)\mapsto \langle f,g\...
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  • 36.4k
2 votes

Solve the ordinary differential equation $\frac{d^2}{dx^2}F(x)=\frac{1}{F(x)^2}$

It is a VERY difficult equation. If you are just interested for a solution try this by Wolfram:
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2 votes

Root in $(1,2]$ of Equation $x^n-x-n=0$

An alternative approach: let $p_n(x)=x^n-x-n$. We may notice that $$ p_n\left(1+\frac{\log(n+1)}{n}\right)=\left(1+\frac{\log(n+1)}{n}\right)^n-(n+1)-\frac{\log(n+1)}{n} < -\frac{\log(n+1)}{n} $$ ...
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2 votes

Convergence of $\sum \frac{b_n}{n}$ Where $b_n = 1, -1, -1, 1, 1, 1, -1, -1,-1,-1,1,1,1,1,1,....$

The $n$-th "run" consists of $n$ elements (numerators of $1$ for odd $n$ and numerators of $-1$ for even $n$). The last element of the $n$-th run has index $n(n+1)/2$, and the first element ...
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2 votes

Ratio of heights of a sphere,over and under water

The volume of a spherical cap is given by the formula (a formula worth remembering) $ V_{cap} = \dfrac{1}{3} \pi h^2 (3 R - h) $ While the total volume of the sphere is $ V_{sphere} = \dfrac{4}{3} \pi ...
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2 votes

Maximum value of the function $\frac{1}{2}x^2(k-x)$ with $x \in [0,k]$

Your approach works fine. Alternatively, observe that: $f(x) = 2\cdot\dfrac{x}{2}\cdot \dfrac{x}{2}\cdot (k-x)\le 2\left(\dfrac{\dfrac{x}{2}+\dfrac{x}{2}+(k-x)}{3}\right)^3= \dfrac{2k^3}{27}$, by AM-...
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  • 3,285
1 vote

notation when changing variables in Partial Differential Equation

Defining $g$ as you have done is indeed the right thing to do, but for the sake of everyone's sanity, might I suggest that the function $h$ be defined as in the order $h(t,r,C)$ (so compose your $h$ ...
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  • 36.4k
1 vote

Assume $\int_a^b (f-\lambda g)^2=0$. Then $\lambda=\frac{\int_a^b fg}{\int_a^b g^2}$. Sub $\lambda$ into first equation. How do we know it's zero?

Let's start with a simpler example $$y(x)=x^2+bx+c=0\tag{1}$$ $$\Delta=b^2-4c$$ Assume $\Delta=0 \implies b^2=4c$. Then $x=\frac{-b}{2}$ is the solution to $(1)$. Note that this solution relies on our ...
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  • 2,057
1 vote

Computing $\int_{0}^{1} \frac{x^{t}-1}{\ln x} d x$ without Feynman, where $Re(t)>-1$?

Using special functions $$I=\int\frac{x^{t}-1}{\log( x)}\,d x=\int\frac{e^{(t+1) y}}{y}\,dy-\int\frac{e^y}{y}\,dy=\text{Ei}((t+1) y)-\text{Ei}(y)=\text{Ei}((t+1) \log (x))-\text{li}(x)$$ $$\lim_{x\to ...
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1 vote

Optimize $xyz$ where $x+y+z=1$ and $x^2+y^2+z^2=1$?

Notice that $$1=1^2=(x+y+z)^2=x^2+y^2+z^2+2(xy+yz+xy)=1+2(xy+yz+xy)$$ Solving for the products, we learn all elementary symmetric polynomials in $(x,y,z)$: \begin{align*} x+y+z&=1 \\ xy+xz+yz&=...
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1 vote

Convergence of $\sum_{n=1}^\infty \frac{\sum_{i=1}^n\frac{1}{ \sqrt i}}{n^2}$

Using generalized harmonic numbers, you face $$S_p=\sum_{n=1}^p \frac{H_n^{\left(\frac{1}{2}\right)}}{n^2}$$ For large values of $n$ $$H_n^{\left(\frac{1}{2}\right)}=2\sqrt{n}+\zeta \left(\frac{1}{2}\...
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1 vote

Convergence of $\sum_{n=1}^\infty \frac{\sum_{i=1}^n\frac{1}{ \sqrt i}}{n^2}$

Just for fun. By the Hermite-Hadamard inequality $$ \sqrt{4n+2}-\sqrt{6}=\int_{3/2}^{n+1/2}\frac{dx}{\sqrt{x}}\geq \sum_{k=2}^{n}\frac{1}{\sqrt{k}} $$ and by the inequalities fulfilled by the central ...
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1 vote

What is the difference between a Limit and Derivative?

The idea of a limit is used to compute the derivative. The derivative is really why Calculus was invented. People wanted to say things about "instantaneous" rates of change, like what the ...
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1 vote

Maximum value of the function $\frac{1}{2}x^2(k-x)$ with $x \in [0,k]$

The Extreme value theorem states that if a function is continuous on a closed interval [a,b], then the function must have a maximum and a minimum on the interval. From this let's calculate the ...
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1 vote

Optimize $xyz$ where $x+y+z=1$ and $x^2+y^2+z^2=1$?

The Lagrange multiplier method is effective for this problem. First of all observe that the case $x=y=z$ is inadmissible. By symmetry we may assume that $z\neq x$ and $z\neq y.$ We have to solve for $$...
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1 vote

Convergence of $\sum \frac{b_n}{n}$ Where $b_n = 1, -1, -1, 1, 1, 1, -1, -1,-1,-1,1,1,1,1,1,....$

Let $S_k=b_1+b_2+\ldots +b_k.$ Denote $$u_n={2n(2n-1)\over 2}=n(2n-1)\quad v_n={(2n+1)2n\over 2}=(2n+1)n$$ Then $$S_{u_n}=n,\qquad S_{v_n}=-1$$ Moreover $S_k$ is decreasing for $u_n\le k\le v_n$ and ...
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1 vote

Looking for a direct way to evaluate $\int_0^1\frac{\ln(x)\ln(2+x)}{1+x}dx$

On one hand, one may rewrite the integral as: $$\begin{align} I&=\frac{1}{2}\underbrace{\int_{0}^{1}\frac{\log^2(x)}{1+x}dx}_{\frac{3}{2}\zeta(3)}+\frac{1}{2}\underbrace{\int_{0}^{1}\frac{\log^2(2+...
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