Deleting a node from XmlDocument
Removing nodes from an XmlDocument
$\lim\limits_{n \to \infty} \frac{a_n}{b_n}=l$ ; Prove $\sum\limits_{k=1}^\infty a_k$ converges iff $\sum\limits_{k=1}^\infty b_k$ converges. Written, in Ex.8 Ch.9.1 of the book Advanced Calculus by P. M. Fitzpatrick :     Suppose that $\sum\limits_{k=1}^\infty a_k$ and $\sum\limits_{k=1}^\infty b_k$ are series of positive numbers such that $$\lim_{n \to \infty} \frac{a_n}{b_n}=l \ \ \ \text{and} \ l&gt;0.$$ Prove that the series $\sum\limits_{k=1}^\infty a_k$ converges iff the series $\sum\limits_{k=1}^\infty b_k$ converges.   Am I correct by the following (sketch of) proof? :  1- For a given $\epsilon_1$ there is $N_1$ such that $\left|\frac{a_n}{b_n} - l\right| &lt; \epsilon_1$ for all $n \ge N_1$.  2- Since the series $\sum\limits_{k=1}^\infty a_k$ converges, for a given $\epsilon_2$ there is $N_2$ such that $\left|b_{n+1}+\dots+b_{n+k}\right|&lt; \epsilon_1$ for all $n \ge N_2$ any for all natural numbers $k$.  3- Define $N = \max {\{N_1,N_2}\}$.   4- From Step (1), $a_{n+k} &lt; (\epsilon_1+l) b_{n+k}$ for all $n \ge N$ any for all natural numbers $k$. [Also, $a_i$'s and $b_i$'s are all positive]. Thus $a_{n+1}+\dots+a_{n+k} &lt; (\epsilon_1+l) (b_{n+1}+\dots+b_{n+k})&lt; \epsilon_3$. Then the convergence of the series $\sum\limits_{k=1}^\infty b_k$ implies the convergence of the series $\sum\limits_{k=1}^\infty a_k$.  5- For the reverse implication, we use the fact that $\lim\limits_{n \to \infty}\frac{a_n}{b_n}=l \iff \lim\limits_{n \to \infty}\frac{b_n}{a_n}= \frac{1}{l}= l' &gt;0$ and repeat the process this time for a and b exchanged.   6- The Quotient Property For Sequences hold for a nonzero limit in the denominator, but since the limit in the numerator also is zero so we may use the The Quotient Property For Sequences.    Thanks. Where do AirDrop files go when I transfer them between an iPad, iPhone and iPod? I AirDropped some Google Docs from my iPad to my iPhone but nothing happened; where did my files go? The file was a list of tools. Favorite questions in the Android app What's the deal with the phrase "could care less"?  Whilst growing up, I've always known people (parents etc) to use the phrase "couldn't care less", but I've also come across people who use the phrase "could care less" to mean the same thing (that is, "I do not care at all about that").  To me "couldn't care less" seems a lot more logical. If I couldn't care less about something, that implies I do not care about it at all. In contrast to this, to my mind the phrase "could care less" seems to indicate "I do care about this to some degree at least", but that's definitely not the meaning intended.  What's going on here? Is my (logical) analysis correct or in error? Free visual similarity image for a local hard drive search? I'm looking for a desktop search engine for finding images on my local hard drive with high visual similarity. I've only found paid for ones so far. It could be Linux or windows one, either would be helpful.  A Google desktop plug-in would be nice! Install pandas to python3.5 instead of 2.7 Can you take different archetype options at different levels? Once you've picked a subclass option at level 3, could you later take a different subclass option, or do you lock into a specialty at level 3 and then run with it for the rest of that class' progression?  For example, as a rogue, if you took the "Thief" archetype at level 3, giving you access to Fast Hands and Second Story Work, could you then take the "Assassin" archetype to give you the specific benefits that archetype gives instead of the "Thief" ones?  The  seems to imply that you couldn't:     At 3rd level, you choose an archetype that you emulate in the exercise of your rogue abilities: Thief, detailed at the end of the class description, or one from another source. Your archetype choice grants you features at 3rd level and then again at 9th, 13th, and 17th level.   but I'd like a more certain answer, especially one that refers to archetypes in general. Also I'm not ashamed to admit I'd prefer a source with some authority that tells me you can mix and match, but I am looking for the truth of the rules when all is said and done...   Different to  because the no on that demonstrates that you can't re-take a level of the same class. Taking different features for different levels is not the same thing, and would work differently. Similar to  but that one was specific to monks - which is why it didn't show up on any searches I did, and why it wouldn't on the searches of others. This question is specific enough to be clear and precise, but broad enough to be useful to others who aren't in the exact same situation/using the same class. What is meant by immutable? This could be the dumbest question ever asked but I think it is quite confusing for a Java newbie.    Can somebody clarify what is meant by immutable?  Why is a String immutable? What are the advantages/disadvantages of the immutable objects? Why should a mutable object such as StringBuilder be preferred over String and vice-verse?   A nice example (in Java) will be really appreciated. 'Which', 'whose' or something else? Setting the editing style on a UITableViewCell How to make a surface with a transparent background in pygame Windows 8 skype is terrible.  They've removed many buttons, most of the preferences, and have in general made it impossibly oversimplified and difficult to use.  Old Skype was fine when I had windows 7.  Is there any way to get it back on Windows 8? GCD of the already GCD Fix for Hibernate error "Use of the same entity name twice" How you fix the following Hibernate error:  What does "Use of the same entity name twice". Genetic linkage greater than 50 centimorgans Classically, the linkage between two loci can be measured in centimorgans (cM), which represents the percent chance that these two loci will recombine an odd number of times (generating a recombinant genotype).  Due to independent assortment, markers on different chromosomes are expected to recombine 50% of the time. As I understand, if one conducts a classic experiment and counts various progeny, then deduces the apparent linkage, values of 50 cM or more are interpreted as "impossible to determine whether on separate chromosomes or just recombines very often". See (), with grammar corrected:     The final point that we need to make regards the maximum distance that we can measure. Because of the way in which the calculations are performed, we can never have more (than) 50% recombinant gametes. Therefore the (maximum) distance that two genes can be apart and still measure that distance is just less (than) 50 cM. If two genes are greater than 50 cM apart, then we can not determine if they reside on the same chromosome or are on different chromosomes.   Wikipedia gives an  and also remarks that (d is physical distance):     The probability of recombination is approximately d/100 for small values of d and approaches 50% as d goes to infinity.   However, what happens if the linkage is greater than 50 cM? It seems like classical experiments aside, such a situation is possible in reality.  For instance, , and . Let's take two genes on Chr.IV:    is at about +631 kb.  is at about +1370 kb.   The physical distance between these is about 734 kb. According to the cM/kbp value, the centimorgan distance is $734\text{ kb} \cdot 0.31\text{ cM}/\text{kb} = 228 \text{ cM}$ . How to interpret this 228?  If I mate a haploid yeast which is DNF2 TOM1 with another which is dnf2 tom1 (lowercase indicating minor allele rather than deletion), then sporulate them, what is the chance of getting spores with DNF2 tom1 or dnf2 TOM1 genotypes?  I realize that the cM/kbp value is only a simplification and that in practice linkage is a more complex phenomenon. Nevertheless, it seems plausible that several crossover events can happen on Chr.IV since it is so big. This includes the possibility of 1, 3, 5 and more crossovers which would create a hybrid (assuming they all happen between these two loci), as well as the possibility of 2, 4, etc crossovers which would create a non-hybrid spore (at least as far as our chosen markers are concerned). USB stick mysteriously become write protected So, somehow over the weekend, my 1GB USB flash drive has managed to become write-protected.  There is no switch on the stick, so I'm concluding that something has corrupted at some point.  Obviously I can't format it or remove.  The output from dmesg | tail is:  Buffer I/O error on device sdb1, logical block 328168 lost page write due to I/O error on sdb1 sd 73:0:0:0: [sdb] Unhandled sense code sd 73:0:0:0: [sdb] Result: hostbyte=DID_OK driverbyte=DRIVER_SENSE sd 73:0:0:0: [sdb] Sense Key : Data Protect [current] Info fld=0x0 sd 73:0:0:0: [sdb] Add. Sense: Write protected sd 73:0:0:0: [sdb] CDB: Write(10): 2a 00 00 05 08 00 00 00 01 00 Buffer I/O error on device sdb1, logical block 328200 lost page write due to I/O error on sdb1   I think the second and final lines are a giveaway to corruption, but I don't know how to interpret this output.  Could someone help to see if this output gives any indication?  I usually eject the drive before removing it, but I may have been lax once or twice.  The data isn't critical, it's just bugging me.  Any suggestions on how to fix it (if the drive even can be) would be greatly appreciated. For every $p\in [1,\infty)$, consider the set $\ell^p$ of real sequences $x=(x_n)=(x_1,x_2,\ldots)$ such that $\sum_{n=1}^\infty |x_n|^p&lt;\infty$. This is a Banach space with the norm $\Vert x\Vert_p=(\sum|x_n|^p)^{1/p}$.  I have the following problem:     Is it true that $\ell^p=\bigcup_{1\leq q&lt;p}\ell^q$ for $p&gt;1$?   I don't think it is true, but I'm having problems to show this. Clearly, the statement is true for some $p&gt;1$ iff it is true for every $p&gt;1$.  I'm trying to find a sequence $(a_n)$ such that $\sum |a_n|^s$ converges iff $s\geq 1$. I've tried many sequences (such as $(n^{1+1/n})$, involving $\log$, etc...), but none of them worked. Is the prefix "pre-" meaningless in the terms "pre-heated" and "pre-board"? This question poses a paradox of meaning. The general question is whether, if two sentences (x and y) can be used in the same situation, with the same literal meaning, and x and y only differ in that x has morphological affixes that y doesn't, aren't the surplus affixes "meaningless" by the alternation test?   Take this example:  I put the chicken inside a preheated oven.    Wouldn't it make more sense to say:  I put the chicken inside a heated oven.      There is only two states an oven can possibly be in—heated or unheated.      Another example: Does first-class board the plane, or pre-board the plane?  What does that mean—to get on before you get on? Can a sequence which decays more slowly still yield a converging series? In Bergman's companion notes to Rudin, he says that "If a sequence of positive terms has convergent sum, so does every sequence of positive terms which decays more rapidly." So given a sequence $\{a_n\}$ of positive terms such that $\sum_n a_n$ converges, if $\{b_n\}$ is such that $$ \lim_{n\to\infty} \frac{a_n}{b_n} = +\infty, $$ then $\sum_nb_n$ converges. I can prove this given $\{a_n\}$ and $\{b_n\}$. However, can we find $b_n$ which decays more slowly, i.e. $$ \lim_{n\to\infty} \frac{b_n}{a_n} = +\infty $$ such that $\sum_n b_n$ converges?  Similarly, we have the claim "if a sequence of positive terms has divergent sum, then so does every sequence of positive terms which decays more slowly." How do I counter-knife an enemy? When you press down the right stick to counter the attack, do you just press it, repeatedly press it, or press and hold? I've been knifed a few times from the front and each time I try to counter I fail. Also, if you believe you are about to be knifed and hold down the right stick until you are knifed, will that counter an attack, or do you have to press it down just at the right time?