## Is there any equivalent of Taylor/Maclaurin series of $ln(1+x)$ for $|x| > 1$

I happened to come across Taylor series and Maclaurin series recently, but everywhere I read about the expansion for $$ln(1+x)$$, it was stated that the approximation is valid for $$-1 < x < 1$$.

I understand that the bounds for $$x$$ are because the series doesn’t converge for $$|x|>1$$, but is there any equivalent of this series for the value of $$|x|$$ as greater than $$1$$?

Please note that I am not asking if we can compute for $$|x|>1$$ or not, as that can be done by computing for $$\frac 1x$$, which will then lie between $$-1$$ and $$1$$.

Also, I’m quite new to all this, so new that today was the day I read the name ‘Maclaurin’ for the first time. So any answers understandable with high school mathematics are highly appreciated.

Thanks!

## $M \leq X$. Is it true that $M^*$ is a subspace of $X^*$ where $*$ denotes the dual space?

Let $$X$$ be a Banach space and $$M$$ be a closed subspace i.e. $$M \leq X$$.

Is it true that $$M^*$$ is a subspace of $$X^*$$ where $$*$$ denotes the dual space?

What I have tried:

I know by Hahn Banach theorem that every $$f \in M^*$$ has an extension $$\hat f \in X^*$$ such that $$\hat f |_M =f$$ and the operator norm doesn’t change i.e. $$\|\hat f\| =\|f\|$$.

I found that such an extension can be chosen to be linear if $$X$$ is a Hilbert space here, hence my claim might not hold in the general case.

Any help is appreciated.

## Let $X$ be a complete metric space. Suppose $\{x_n\}$ is a sequence in $X$…

Let $$X$$ be a complete metric space. Suppose $$\{x_n\}$$ is a sequence in $$X$$ with the property that $$|x_n – x_m| < \frac{1}{n}+\frac{1}{m}$$ for all $$n, m \in \mathbb{N}$$. Show that $$\{x_n\}$$ is convergent.

I’m struggling to find where to start for this problem and would appreciate some help.

## Find harmonic of order $3$ for $2\pi$ periodic funciton $f(x) = \pi -|x|$

Let $$f$$ be a $$2\pi$$ periodic function such that $$f(x) = \pi -|x|$$ in $$[-\pi,\pi]$$. Find its harmonic of order $$3$$

I’m learning about orthogonal families of polynmials and discrete polynomial approximations by fourier functions. I’ve came upon this exercise, but I don’t know what an harmonic of order $$3$$ means.

## What will be $\lim\limits_{x \rightarrow 0^+} \left ( 2 \sin {\frac {1} {x}} + \sqrt x \sin {\frac {1} {x}} \right )^x.$

Evaluate

$$\lim\limits_{x \rightarrow 0^+} \left ( 2 \sin {\frac {1} {x}} + \sqrt x \sin {\frac {1} {x}} \right )^x.$$

I tried by taking log but it wouldn’t work because there are infinitely many points in any neighbourhood of $$0$$ where $$\ln \left ( \sin {\frac {1} {x}} \right )$$ doesn’t exist. How to overcome this situation?

Any help will be highly appreciated.Thank you very much for your valuable time.

## Given an algebra structure $(X,*)$ s.t. $(x*y)*y = y*(y*x) = x$ , prove$x*y=y*x$.

Suppose $$(X,*)$$ is arbitrary algebraic structure such that $$\forall x,y\in X$$, we have $$(x*y)*y = y*(y*x) = x$$, prove that $$x*y=y*x$$.

This question seems pretty simple but I tried and I failed.

## How to update the apt-check count of “X packages can be updated” message

Is there a command to update the apt-check results so I can use bash script to check them?

I run upgrade and it shows 0 but apt-check still shows 12 packages can be updated. How often is this suppose to update on its own? I have numerous servers. Some seem to update instantly. Others go days still showing updates are available when they aren’t.

I have read that rebooting can reset it but thats silly. Why would you need to reboot to refresh a simple message especially when most updates do NOT require a reboot.

# apt-get update && apt-get upgrade 0 upgraded, 0 newly installed, 0 to remove # /usr/lib/update-notifier/apt-check --human-readable 12 packages can be updated. 9 updates are security updates.