C# How to fix DataGridView Cross-thread operation not valid
Cross-thread operation not valid: Control accessed from a thread other than the thread it was created on
Proving the geometric sum formula by induction $$\sum_{k=0}^nq^k = \frac{1-q^{n+1}}{1-q}$$  I want to prove this by induction. Here's what I have.  $$\frac{1-q^{n+1}}{1-q} + q^{n+1} = \frac{1-q^{n+1}+q^{n+1}(1-q)}{1-q}$$  I wanted to factor a $q^{n+1}$ out of the second expression but that 1- is screwing it up... How can I log into a different Google account? It only shows one of my Google accounts. I want to log into a different one but I don't want them connected in any way. What do I do? What are the meaning and possible origin of "word!" and "word up"? Several times, I have had conversations, all over instant messenger, finish with "word" or "Word up G".  As it ends a conversation, I am guessing it is like "goodbye".  My question is what is the meaning of "word" and "word up g"? Also, what is the origin?  I am more interested in the meaning as that will help with understanding its usage. What are the technical advantages and disadvantages of mirrorless? The main doubt one can have about mirrorless is, I think that they do not provide ttl viewfinder. Is this a big disadvantage? In what way is it worst (or best) than classical reflex system?  What are the others advantages and disadvantages of a mirrorless compared to a reflex?  Have mirrorless a chance to replace reflex (at least for non pros) in the future? How do I root my Android device? This is a common question asked by those who want to root their phones. How exactly do I root my Android device? Are there risks involved? How exactly does Knuth's Up-Arrow notation work? I'm having trouble with a proof by induction Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$ Suppose $f$ and $g$ are continuous on $[a,b]$ and that $f(a) &lt; g(a)$ but $f(b) &gt; g(b)$.   Prove that $f(x) = g(x)$ for some $x \in [a,b]$.     I did try and actually did get as far as coming up with $f-g$. Would I set $h(x)= f(x)- g(x)$ and say $h(a) &lt;x&lt; h(b)$? Standard nomenclature for component types What prefixes should be used on reference designators for components of various types?  I think we can all agree that "R##" refers to a resistor, "C##" refers to a capacitor, and "L##" refers to an inductor.  Beyond that, it appears to be a mishmash of conventions.    I've seen both "IC" and "U" used for chips.   "Q", "T", and "M" have been used for transistors and MOSFETs.  Ordinarily, "D##" is used for diodes, but LEDs and Zeners get special treatment on some boards.    What standards are available for reference, and who endorses them?  JEDEC, ISO, IEEE, and other standardization bodies are welcome references (though I'd prefer an inexpensive standard), but I'm also curious to see what's used at various companies other than my own.    Edit:  What I'd really like to see is a list in an answer here which complies with the standard (even if it's just a quote). No splitter widget in new separate window I am new to Blender and I am trying to find my way through its interface. I want to have several windows (I mean OS windows, not Blender windows). So I create a separate window by dragging splitter widget while holding Shift.  The problem is there are no splitter widgets in the newly created OS window. So I can't have multiple blender windows in it. The new window does not have a Layout type drop down menu either. So how do I add new Blender windows in a new OS window? (Sorry if I'm using the wrong terms, but hopefully its clear what I am trying to achieve). How to show that $f_{0}f_{1}+f_{1}f_{2}+...+f_{2n-1}f_{2n} = f^{2}_{2n}$ The following problem:  The Fibonacci number is denoted by  $f_{n}$, show that the following holds when $n$ is a positive integer:  $f_{0}f_{1}+f_{1}f_{2}+...+f_{2n-1}f_{2n} = f^{2}_{2n}$  My idea was to do an induction proof, which holds for $n$ = 1  $f_{0}f_{1} + f_{1}f_{2} = f^{2}_{2}$  So we assume it holds for $n$ and show for $n+1$  $f_{0}f_{1}+f_{1}f_{2}+...+f_{2n-1}f_{2n} + f_{2n}f_{2n+1} = f^{2}_{2n} + f_{2n}f_{2n+1}$  I fail to get  $f^{2}_{2n+1}$ out of the term  $f^{2}_{2n} + f_{2n}f_{2n+1}$  Is there a difference between "show that" and "prove that". So might be induction the wrong tool here?  Many thanks for your hints in advance :-) Java - Could not find the main class What are the consequences of using a low-wattage USB-C power supply with a MacBook Pro? My wife was just issued a multiple entry visa by France, which is good for one year.  Our first trip will be to France, the issuing country.  My question is: after we leave the EU and return, must future trips also start in France? Or may we go directly to (say) Greece from the US on a later visit? Wifi problems with rtl8723be in Ubuntu 14.04 How to add field history and product as a related list of opportunity in vf page I am trying the below code to add field history and product as a related list of opportunity in vf page, but its not working .  &lt;apex:relatedList list="OpportunityFieldHistory"/&gt; &lt;apex:relatedList list="Product"/&gt;   Can anyone tell me what is the correct way to write it.  Regards    Edit:  There are two Standard objects that have Opportunity history information,  and . The relatedList component works fine with a value of OpportunityHistories for the list attribute, but an error is thrown when using the value of Histories, even though the SObject describe information shows "Histories" as the child relationship name.   &lt;apex:relatedList list="OpportunityHistories"/&gt; &lt;!-- Works fine --&gt; &lt;apex:relatedList list="Histories"/&gt;            &lt;!-- Error --&gt; How to get the native version of Spotify running? Integral closure of $\mathbb{Z}$ in $\mathbb{C}$ is not finitely generated as a $\mathbb{Z}$-module? Let     $$ \mathbb{Z}^{'}_{\mathbb{C}}=\{ z \in \mathbb{C} | \exists f \in \mathbb{Z}[X] \text{ monic such that } f(z)=0\} $$   be the integral closure of $ \mathbb{Z} $ in $ \mathbb{C} $. Prove that $ \mathbb{Z}^{'}_{\mathbb{C}} $ is not finitely generated as a $ \mathbb{Z} $-module.  My attempt: Suppose $ \mathbb{Z}^{'}_{\mathbb{C}}=\mathbb{Z}\alpha_{1}+...+\mathbb{Z}\alpha_{n} $ for some $ \{ \alpha_{i}\}_{i=\overline{1,n}} \subset \mathbb{Z}^{'}_{\mathbb{C}} $. I know that given $ \alpha , \beta \in \mathbb{Z}^{'}_{\mathbb{C}} $ with minimal polynomials of degrees $ m $ and $ n $ respectively, then we can construct a monic polynomial with integral coefficients $ P(z)=\prod(z-(\alpha_{i}+\beta_{j})) $ where $ \alpha_{i},\beta_{j} $ are the conjugates of $\alpha $ , $\beta $ and therefore the minimal polynomial of $ \alpha + \beta $ has degree $ \leq mn $.   I was thinking that then the degrees of the minimal polynomials of the elements of $ \mathbb{Z}\alpha_{1}+...\mathbb{Z}\alpha_{n} $ could be somehow bounded by a certain constant, whereas $ \mathbb{Z}^{'}_{\mathbb{C}}$ contains elements whose degrees of the minimal polynomials are arbitrarily large, take $ 2^{\frac{1}{n}} $ for example,$ n\geq 2$. I think this works when we are asked to prove that the set of all algebraic numbers $ \overline{\mathbb{Q}} $ is not a finitely dimensional $\mathbb{Q}$-vector space, but I don't know how to proceed in my case.  I would really appreciate any ideas, hints or solutions. Thank you very much! URLs: Should I use hyphens, underscores or plus symbols? Which is better for search engines?   example.com/my_cool_page.html example.com/my-cool-page.html  example.com/my+cool+page.html In general, when can I omit that from a sentence?   Can I omit that (emphasized) in sentence (2) below?        We say that such algorithms handle concept drift and can learn from time-changing data streams.   When the statistical properties of the target variable change, we say that concept drift has occurred.   She told me that some of her classmates failed.      To me, sentences (1) and (3) sound better without that. If I omit that from (2), it just doesn't sound right when I read it out loud. (I should point out I'm not a native speaker.)  How do you decide when to omit it? Is it just a matter of taste? (I know that is sometimes mandatory.)  Can you think of sentences that "sound better" with that? Can you think of sentences that "sound better" without that? Can you think of a few sentences where it is wrong to omit that?