(whenever $n$ is a non-negative integer)   I did the basic step $P(1)$ and found the statment $P(n+1)$  I now have $(n+1)! - 1 + (n+1)\cdot(n+1)!$  This should equal $(n+2)! - 1$, but how do I show that?
I am trying to prove the following by Mathematical Induction:  $$\sum_{i=1}^n i\cdot i! = (n+1)!-1\quad\text{for all integers $n\ge 1$}$$   My proof by Induction follows: First prove $P(1)$ is true,  $$\sum_{i=1}^1 i\cdot i! = (1+1)! - 1$$  Then, for all integers $k &gt;= 1$, if $P(k)$ is true then $P(k+1)$ is also true,  $$ \sum_{i=1}^k i\cdot i! = (k+1)! - 1$$ Then, we must show that $P(k+1)$ is true   $$ \sum_{i=1}^{k+1} i\cdot i! = ((k+1)+1)! - 1\\ \sum_{i=1}^{k+1} i\cdot i! = (k+2)! - 1$$  I am currently struggling with the next step to show that $P(k)$ is equal to $P(k+1)$. How would I finish this prove by induction?
Calculate $\sum_{k=1}^n (-1)^{k+1} \binom{n}{k}\frac{1}{k}$ Are there critical hits for spells? Save text area to file Writing someone else's master's thesis: Unethical and illegal? It is really difficult for me to decide on this situation: when someone calls me and asks for work-for-hire for his/her thesis. I don't know what to do. I want to help and earn money but I think this way of earning money is unethical.  I think that receiving the title of thesis and giving back prepared manuscript is the intention that some people have, when asking for their thesis to be written.  There are plenty of websites offering such services!   The question is that: Is it legal and or ethical to write someone else's master's thesis as work-for-hire? Are there any rules for adjudicating a ranged attack fired at a target engaged in melee combat with an ally? While running a session recently, one of the players chose to "assist" his companion who was in melee with a goblin, by firing an arrow at the pair of them. I couldn't find any particular rules for this scenario so just had him roll at a disadvantage, counting a miss as a hit on his companion (damage to be ascertained by a second roll).   This seemed to work okay, and became a running joke for the night as the arrow bounced off the companions back, but are there specific rules for this scenario? At the end of the game, both remaining players have a Baron.   The sum of all cards discarded by both players is equal.  Who wins? How TDD-oriented is the SE team? Stack Exchange is my role-model for application development.  I've tried to learn as much as possible from the blog and podcasts, from thought process to design, architecture, convention, etc etc.  I'm just about to wrap my head around TDD and am curious about how the SE team approaches it.  Is there a methodology that all team members follow, or is it more developer-driven testing where you pick complex bits to unit test as needed, or is it more about catch-all integration testing once a piece of functionality is done, or... ?  Any input would be fantastic. Why does du and quota results not match? What adapter to mount a screw thread lens on Nikon D90? I have a Soligor 450 mm lens with a 40 mm dia camera body end thread connection.  I would like to try adapting it to a D90 AF system.  Are there adapter rings available out there for this purpose? Now that I have Google Chrome all set up, I got myself a new laptop. Does anyone have any idea how to transfer my settings, cookies, favorites, etc?   I checked documents and settings\me\application data, and found nothing. What is the difference between == and equals() in Java? 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The resulting image:    \documentclass{standalone} \setlength\PreviewBorder{5pt}%  \usepackage{tikz}  \newcommand*{\Width}{2.0} \newcommand*{\Height}{3.0} \newcommand*{\Depth}{1.0}  \begin{document} \begin{tikzpicture}%     \coordinate (A) at ( 0,       0,     \Depth);%     \coordinate (B) at (\Height,  0,     \Depth);%     \coordinate (C) at (\Height,  0,      0);%     \coordinate (D) at (\Height, \Width,  0);%     \coordinate (E) at ( 0,      \Width,  0);%     \coordinate (F) at ( 0,      \Width, \Depth);%     \coordinate (G) at (\Height, \Width, \Depth);%     \coordinate (H) at ( 0,       0,      0);%     %     \draw [blue, ultra thick,line cap=rect]%         (A) -- (B) -- (C) -- (D) -- (E) -- (F) -- cycle;%     \draw [blue, ultra thick,line cap=rect]%         (B) -- (G) -- (F) -- (E) -- (D) -- (G) -- cycle;%      \path (D)  [right] node {$d$};% Problem Point 1     \path (F)  [left ] node {$f$};% Problem Point 2 \end{tikzpicture}% \end{document}   This problem about  seems related, but I do not understand how to adapt that to my problem. PHP errors NOT being displayed in the browser [Ubuntu 10.10] Creating queries to select by attribute with arcpy I'm quite new with arcpy and I'm finding selecting by attribute a bit of a nightmare. There's a problem with my query on the script below [query = "\'Location\' = observer"], but I can't tell what it is. 'Location' is a field of the feature class I'm dealing with (fc_copy).  with arcpy.da.UpdateCursor(fc_copy,field_names) as cursor:        for row in cursor:         observer = row[0]         arcpy.MakeFeatureLayer_management(fc_copy, 'fl_copy_OBS')         arcpy.MakeFeatureLayer_management(fc_copy, 'fl_copy_TAR')         query = "\'Location\' = observer"         arcpy.SelectLayerByAttribute_management('fl_copy_OBS', 'NEW_SELECTION', query)         arcpy.SelectLayerByAttribute_management('fl_copy_TAR', 'NEW_SELECTION', query, invert_where_clause = 'INVERT')   This is the error I get => Invalid expression                            Failed to execute (SelectLayerByAttribute).  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It appears I only have the free month that came with my Battlechest purchase.  I just created and bought my account this morning. Would this affect the free referral month in any way?    We haven't seen each other or partied online in the game yet, do we need to do that first in order for me to receive the month? Is there a "do ... while" loop in Ruby? I'm using this code to let the user enter in names while the program stores them in an array until they enter an empty string (they must press enter after each name):  people = [] info = 'a' # must fill variable with something, otherwise loop won't execute  while not info.empty?     info = gets.chomp     people += [Person.new(info)] if not info.empty? end   This code would look much nicer in a do ... while loop:  people = []  do     info = gets.chomp     people += [Person.new(info)] if not info.empty? while not info.empty?   In this code I don't have to assign info to some random string.  Unfortunately this type of loop doesn't seem to exist in Ruby. Can anybody suggest a better way of doing this? File upload with Javascript without user intervention