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Tagged with logarithms taylor-expansion
149
questions
2
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Prove $\ln (1+x) \leq x - x^2/4 $ for $x \leq 1$ using Taylor's theorem
RTP: $\ln (1+x) \leq x - \frac{1}{4} x^2$ for $x \leq 1$
I am trying to prove this specifically using Taylor theorem. Here is what I have so far:
$\ln (1+x) = x - \frac{x^2}{2} + \frac{x^3}{3(1+\xi)^3}...
- 487
3
votes
1
answer
82
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Showing the quintessential logarithm property using the Maclaurin series of $\log$
For $-1\le x<1$, we have
$$\log(1-x) = -\sum_{k=1}^{\infty} \frac{x^{k}}{k}\\$$
Taking $a,b$ with $|a|,|b|<1$ and $(1-a)(1-b)\le2$, on the function side clearly we have
$$\log(1-a)+\log(1-b) = \...
- 8,369
0
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0
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51
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ln n and pi breakup sequence
To compute the logarithm of an integer efficiently using pen and paper, one can first write the number as a product of numbers close to $1$. For example, $$10 = \left(\frac{64}{63}\right)^{63}\left(\...
- 303
2
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0
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Show that $ \lim_{x \to -\infty} (1 + \sum_{n=1}^{\infty} (\frac{x}{\ln^2(n+1)})^n ) = 0$
Let $x$ be real and define the entire function $f(x)$ as
$$ f(x) = 1 + \sum_{n=1}^{\infty} (\frac{x}{\ln^2(n+1)})^n $$
Now we have that
$$ \lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} (1 + \sum_{...
- 16.4k
3
votes
1
answer
112
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verifying the Taylor expansion of ln(1+x) satisfies the properties of logarithm
Background:
I want to verify the Taylor expansion of $\ln(1+x)$ does satisfy the properties of logarithm.
Question:
Let $f(1+x) = x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots$. Without using the ...
- 127
0
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0
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151
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What will be the series for $ \ln(1 + x) $ where $ x > 1 \in \mathbb{R} $? [duplicate]
I'm developing my own Real Number class in Python for my projects. In that class, user can store or access $ n \in \mathbb{N} $ number of digits after the decimal point.
Everything is completed, ...
0
votes
1
answer
101
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Write the function $\log(1+x), x\in(-1,1]$ by expanding it into an infinite series by application of Taylor's Theorem.
Write the function $\log(1+x), x\in(-1,1]$ by expanding it into an infinite series by application of Taylor's Theorem.
I tried solving this as follows:
We know that,
(Taylor's Theorem in Cauchy's form ...
- 3,067
1
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1
answer
69
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$O(\exp(\ln(x) \ln(\ln(x))^2)) = \sum_{n=0}^{\infty} a_n x^n$ and $0 < a_n$ asymptotics?
Consider for $x>e^e$ and $0<A<5$:
$$g_A(x) = \exp(\ln(x) \ln(\ln(x))^A)$$
I want to find a function $f_A(x)$ ,
$$f_A(x) = a_0(A) + a_1(A) x + a_2(A) x^2 + ... = \sum_{n=0}^{\infty} a_n(A) x^n$...
- 16.4k
2
votes
1
answer
430
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Closed-form Maclaurin series of $\log\cosh(x)$
I know the first few terms are:
$$
\log\cosh(x) = \frac{x^2}{2} - \frac{x^4}{12} + \frac{x^6}{45} + \mathcal{O}(x^8)
$$
Is there a closed-form expression for the $n$'th coefficient of this expansion?
- 6,771
0
votes
1
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53
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$|\frac{x^h -1}{h}-\log(x)|\leq \frac{|h||\log(x)|^2 }{2}\exp(|h\log(x)|), \quad 0<|h|<1.$
I'm trying to prove the following inequality: $$\left\lvert\frac{x^h -1}{h}-\log(x)\right\rvert\leq \frac{|h||\log(x)|^2 }{2}\exp(|h\log(x)|), \quad 0<|h|<1.$$
What I've tried: By the Taylor ...
- 2,783
0
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0
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45
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Inverse exponential formula
The exponential formula (or polymer expansion, see https://en.wikipedia.org/wiki/Exponential_formula), allows one to transform a power series:
$$f(x) = a_1x+\frac{a_2}{2}x^2 + \dots = \sum_{n=1}^\...
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0
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45
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Approximating difference betwen $\log$ function
How should the difference between $\log$ at different points $a, b$ be approximated (say via the Taylor series)?
So, I'm trying to approximate the following term $\log(a) - \log(b)$ where $a, b \in [0,...
- 1
2
votes
1
answer
66
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Use Taylor's theorem to derive an expression for $\log\left(1-\frac{1}{(n+1)^\alpha}\right)-\log\left(1-\frac{1}{n^\alpha}\right)$
In Some asymptotic expressions in the theory of numbers, the author asserts on p.119 that
$\log\left(1-\frac{1}{(n+1)^\alpha}\right)-\log\left(1-\frac{1}{n^\alpha}\right)=\frac{\alpha}{n^{1+\alpha}-n}-...
3
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0
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Show that $\exp\circ\log = \operatorname{id}$ with Taylor series [duplicate]
Good evening!
We all know that $\exp(x) = \sum\limits_{k=0}^\infty \frac{x^k}{k!}$ holds for all $x\in\mathbb{R}$ and $\log(x) = \sum\limits_{j=1}^\infty (-1)^{j-1}\frac{(x-1)^j}{j}$ holds for $|x-1|&...
- 2,766
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votes
3
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201
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Representing $\frac{1}{x^2}$ in powers of $(x+2)$
I was asked to represent $f(x)=\frac{1}{x^2}$ in powers of $(x+2)$ using the fact that $\frac{1}{1 − x} = 1 + x + x^2 + x^3 + ...$. I am able to represent $\frac{1}{x^2}$ as a power series, but I am ...
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