I know that if $\sum_{n=0}^{\infty} c_n (x-a)^n$ is a power series with radius of convergence $R$ and $f\colon (a-R,a+R)\to\mathbb{R}$ is the function such that $f(x):=\sum_{n=0}^{\infty} c_n (x-a)^n$ then $f$ is differentiable on $(a-R,a+R)$ and for any $0&lt;r&lt;R$, the series $\sum_{n=1}^{\infty} nc_n (x-a)^{n-1}$ converges uniformly to $f'$ on the interval $[a-r,a+r]$.  Now, since $\cos(x)=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}x^{2n}$ and $\sin(x)=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}x^{2n+1}$ we have $\cos'(x)=\sum_{n=1}^{\infty}(2n)\frac{(-1)^n}{(2n)!}x^{2n-1}=\sum_{n=0}^{\infty}(2(n+1))\frac{(-1)^{n+1}}{(2(n+1))!}x^{2(n+1)-1}=-\sum_{n=0}^{\infty} \frac{2n+2}{2n+2}\frac{(-1)^n}{(2n+1)!}x^{2n+1}=-\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}x^{2n+1}=-\sin(x)$  and  $\sin'(x)=\sum_{n=1}^{\infty}(2n+1)\frac{(-1)^n}{(2n+1)!}x^{2n+1-1}=\sum_{n=1}^{\infty}\frac{(-1)^n}{(2n)!}x^{2n}$ and now I'm stuck: I don't know how to recover from here the series of $\cos(x)$.   So, where I'm going wrong?
At  I've read that being $\sin(x)=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}x^{2n+1}$ by using  we get $\sin'(x)=\sum_{n=0}^{\infty}(2n+1) \frac{(-1)^n}{(2n+1)!}x^{2n}=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}x^{2n}=\cos(x)$ but the page  says that if $f(x)=\sum_{n=0}^{\infty}c_n(x-a)^n$ then inside the radius of convergence of the power series we have $f'(x)=\sum_{n=1}^{\infty}nc_n(x-a)^{n-1}$ so shouldn't it be $\sin'(x)=\sum_{n=1}^{\infty}(2n+1)\frac{(-1)^n}{(2n+1)!}x^{2n+1-1}=\sum_{n=1}^{\infty}\frac{(-1)^n}{(2n)!}x^{2n}$?  Now I understand that this last result has a missing term due to the index of the series being $1$ and not $0$ but being the $1$ mandatory because of the above theorem and since trying to shift index doesn't help (we get $\sin'(x)=\sum_{n=0}^{\infty}\frac{(-1)^{n+1}}{(2n+2)!}x^{2n+2}$) how can the result that ProofWiki shows be rigorously justified?   (I mean, I think we can't just say "oh, there's a missing term!" after having applied the theorem and magically shift the index of the series to obtain the desired result)
How Does One Transfer A Steam Game? I have a failure to transfer my game from one HDD to another, and one PC to another. I copied the entire Skyrim folder from one HDD to another. And told where to find it also. Steam did a huge check of my HDD.  But it ended up downloading most of the files anyway. It took a week to download Dawnguard, Hearthfire, and Dragonborn files along with my Hi-Res textures.  I did ask about it in chat. But I feel this could help me or somebody else.  This is the screen I am looking at after transferring my entire hard drive to to new one. Steam downloaded my games, including Skyrim even though my Skyrim data folder had all the data files for Dawnguard, Hearthfire, Hi-Res and Dragonborn.  I have my copied files and have got rid of everything not steam related. I just have my DLCs and they include my hires files.  This is what it looks like.    But I still get this :     What could be causing this? How can I rectify? And in future when one is migrating to a new PC, how can this be avoided?  EDIT::::::::::::::::: Removed everything : Here is how it looks :  Still the prompt as before Steam Error 15  (Side Note : I transferred all my Origin games just by copy/pasting. This really is not nice.)    EDIT 2 :::::::::::::::::::::  And one more EDIT  I tried getting game cache up to speed, but this is prompt I am getting. But I played the game, it was working, I could pick out my character and all, and escape from Helgen.  EDITED 3 :::::::::::::::::  How does one transfer a Steam game? A game that is heavily modded, will the backup a Steam Game option suffice? How do I change the MAC address of my network card? I have a local network in a Windows system where the Admin gives permission to the PCs by their MAC address. But he is on vacations and I have a new PC.  How can I change the MAC address of my new PC to have the same MAC address as the old one?  I know this can be used for spoofing the address, but this is local, so I don't think I will have a problem. Complete list of Lollipop system icons shown in notification bar? Is there a complete list of these? I've seen lots of Q&amp;A about "What does x icon mean" but it would be really useful if there were a complete list for reference! How to colorize an image as in the example? Is there an intelligent way to left-align the parts right of &amp; of the two cases environments shown below (if x >= 0, otherwise, if y &lt; 0, otherwise). I wonder if there is a better way than using manual spaces:  \begin{align*}  u(x) &amp;=   \begin{cases}    0        &amp; \text{if } x \geq 0 \\    1        &amp; \text{otherwise}   \end{cases}   \\  \tau &amp;=   \begin{cases}    a+b+c+d  &amp; \text{if } y &lt; 0 \\    1        &amp; \text{otherwise}   \end{cases} \end{align*} The question is about the encryption equation in asymmetric encryption.  For $c\equiv m^e\bmod n$, prove that for every unique $m$ there is a unique $c$.  Here $n=pq$ where $p$ and $q$ are primes larger than $m$ and the exponent $e$, is $\geq3$ and coprime with $p-1$ and $q-1$.  Thanks! Density of $\mathbb{Q}^n$ in $\mathbb{R}^n$ This was a question from my textbook. This is how I approached it.   For the first part, you can do c(10,5) so you choose 5 of the 10 to be on a team.   For the second part, you have 10 ways to choose the "best player" and you multiply it by c(10,4) to pick the other 4 players. However, I am not sure how you can exclude the "worst player". Although this sounds subjective, I believe the question is saying that the "best" and "worst" players are distinct, whereas the other players aren't. Am I being charged taxes on my grocery purchases at Walmart? Today, I only bought groceries at Walmart and noticed that there is no tax on groceries in California (Google tells me that this is perhaps true for whole of the US?)  So far I have always paid tax at Walmart for my grocery purchases, but that also included items like crackers, shampoo etc  Did Walmart charge me then only for the items that were not groceries, or if your cart has groceries + non-grocery items, you get charged tax on all of them?  Or is this a policy that varies by store, or cashiers' knowledge/discretion, etc?  Would people recommend for me to break down a single mixed purchase into two: one only having groceries and the other with the rest, or is this being paranoid?  My grocery purchases, every two weeks, amount to around $100+.  I pay 8.25% tax, so that is enough to get me free gas if I can figure this out! I know that one can use "they" / "them" / "their" in place of "he" / "him" / "his" or "she" / "her" / "hers" when the subject's / direct object's gender is not known; for instance, just looking at the default user's about me section on Stack Exchange sites:     Apparently, this user prefers to keep an air of mystery about them   My question is what happens when someone wants to reference the subject using a pronoun instead.  Should the verb be conjugated accordingly to the pronoun? E.g.:     Apparently, they prefer to keep an air of mystery about them   Or should the conjugation be retained? E.g.:     Apparently, they prefers to keep an air of mystery about them   The former looks unclear because it wouldn't allow the reader / listener to understand if the writer / speaker is talking about a single person or about a group of people, however the latter sounds very weird, at least just to the non-native speaker which I am. What formula would you use to add up a column based on results from two drop down menus? What is the best regular expression to check if a string is a valid URL? How to disable auto brightening LCD Use a partitioned live usb I have a USB device and i'm trying to create it in a way that it has 2 partitions: one for a live linux disc and the other for document storage.  I created the partitions using gparted and and set a boot flag to the one I want to use as the live disc. Now, I have a usb like this:  Disk /dev/sdc: 14.6 GiB, 15623782400 bytes, 30515200 sectors Units: sectors of 1 * 512 = 512 bytes Sector size (logical/physical): 512 bytes / 512 bytes I/O size (minimum/optimal): 512 bytes / 512 bytes Disklabel type: dos Disk identifier: 0xc3072e18  Device     Boot   Start      End  Sectors  Size Id Type /dev/sdc1       8439808 30515199 22075392 10.5G 83 Linux /dev/sdc2  *      51200  8439807  8388608    4G  b W95 FAT32   I then used dd to flash an Ubuntu iso to /dev/sdc2  sudo dd if=/dev/shm/ubuntu-17.04-desktop-amd64.iso of=/dev/sdc2 bs=4M   When the disc is flashed onto the usb drive, I try to boot from my laptop and it shows "Operating system not found". When i try to use qemu/kvm, it shows a kernel panic like this:    How would I be able to do this properly? How can I specify a long list of math operators? The preambles of my LaTeX documents often include many many lines of \DeclareMathOperator instructions, e.g.  \DeclareMathOperator{\Rep}{Rep} \DeclareMathOperator{\Tet}{Tet} \DeclareMathOperator{\Maps}{Maps} \DeclareMathOperator{\Diff}{Diff}   Is there a nice way to map some macro over a list, so I could do this more concisely, and with less copying and pasting when I add a new math operator? Is my model any good, based on the diagnostic metric ($R^2$/ AUC/ accuracy/ RMSE etc.) value? How to prove that homomorphism from field to ring is injective or zero? Possible Duplicate:          $f:K\rightarrow R$ is a homomorphism from field to ring. To prove is that it is injective or zero. Primitive Pythagorean triple divisible by 3 I have tried a little bit which as follows- Since $f(x_n)\in[0,1]$, $\{f(x_n)\}$ has a convergent subsequence say $y_n=f(x_{r_n})\ \forall n\in\Bbb{N}$ Let, $\lim y_n=l\implies \lim \frac{y_1+y_2+\cdots+y_n}{n}=l\implies \lim \frac{f(x_{r_1})+f(x_{r_2})+\cdots+f(x_{r_n})}{n}=l$ But I am getting no idea to proceed and prove the convergence of $\{x_n\}$. I have also tried to prove the sequence to be cauchy which goes- $x_{m+1}-x_{n+1}={\sum_{i=1}^m f(x_i)\over m}-{\sum_{i=1}^n f(x_i)\over n}\le {\sum_{i=1}^m f(x_i)\over n}-{\sum_{i=1}^n f(x_i)\over n}$ (since $m\ge n)$ $\implies |x_{m+1}-x_{n+1}|\le \frac{|f(x_m)|+|f(x_{m-1})+\cdots+|f(x_{n+1})|}{n}\le\frac{m-n}{n}$ (since $f([0,1])\subseteq [0,1])$ Now, what will I get if I tend $m,n\to\infty$? But I need to do something more in the 2nd case since I have nowhere used the continuity of $f$.  Can anybody give an idea to prove it? Thanks for assistance in advance. 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