I need the proof of sine summation series. Research effort: Seen my notes and the first answer here: math.stackexchange.com/questions/17966/… but it's incomplete and I don't know how to proceed.
How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? For example here is the sum of $\cos$ series:
$$\sum_{k=0}^{n-1}\cos (a+k \cdot d) =\frac{\sin(n \times \frac{d}{2})}{\sin ( \frac{d}{2} )} \times \cos \biggl( \frac{ 2 a + (n-1)\cdot d}{2}\biggr)$$
There ...
@TedShifrin How do I continue after the last step he has shown? He says, "Then by doing the same thing you will have some terms cancelled out. You can easily see which terms are going to get Cancelled. Proceed and you should be able to get the formula."
@Faust: Good advice when you see "proof by induction" — induction on what? for starters.
I'm not going to type all that out, @Abcd, and I have to leave. Write out his formula for $k=0$, for $k=1$, and for $k=2$. Write them above one another.
$Ax^4 +Bx^3 +Cx^2 +Dx +E=0 $ assuming A,B,C,D,E are integers can i use the rational zeros theorem to prove that this equation does or does not have any rational roots?
for some reason i thought none of the rules applied to roots once we broke the $x^3$ barrier
My attempts: $cos x = sin^{1/2}(x)$. Then I substituted this in the given equation and kept solving and solving and solving but couldn't reach any answer.
This question is related to one asked by somebody else here; however, unlike them, I do not want to show that a vector space over a field $F$ and of dimension $n$ is isomorphic to the $n$-th direct power $F^{n}$ of $F$. Instead, I want to generalize this to the following case:
If $B$ is a bas...
The full text of the problem says "Using representation of elements of a direct sum $\oplus_{\alpha \in I}F$ as functions $f: I \to F$, prove the second statement of Proposition 15:18: If $B$ is a basis for a vector space $V$ over $F$, then $V$ is isomorphic to the direct sum $\oplus _{\alpha \in B}F$ of copies of $F$ indexed by $B$."
Proposition 15:18 states the following "A vector space over a field $F$ and of dimension $n$ is isomorphic to an $n$-th direct power $F^{n}$ of $F$. More generally, if $B$ is a basis for a vector space $V$ over $F$, then $V$ is isomorphic to the direct sum $\oplus_{\alpha \in B}F$ of copies of $F$ indexed by $B$".
We defined direct products of groups last semester: The direct product of a family of groups $\{ G_{\alpha}\}_{\alpha \in I}$ is the group $\prod_{\alpha \in I}G_{\alpha}$ with componentwise multiplication as the group operation.
Suppose that $Z[x]$ is composed of all numbers of the form $c + dx$ where $c$ and d are integers and $x$ is an irrational positive solution to the polynomial $x^2 + b = 0$ where $b$ is an integer.
I seek to show that there exist any elements that are units aside from $1$ and $-1$. I know that...
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@TedShifrin i got asked to jion the honours program at my uni this summer and finally caved and said yes. im alittle worried about having to do an honours project and thesis, which is why i have refused until now. (mostly because anything in math thats unsolved seems incredibly difficult to solve.) what exactly will i be doing a project on?
Although you could do some investigative ("baby research") work, most Honors theses people do are just a book report or exposition of some material you've learned in some depth.
abstract algerbra has been my favorite thus far i also really enjoyed some stuff i did on chaotic systems some bifurcations of systems in 3d specifically homoclinical loop bifurcations and im rather interested in topology as well
i dont really like discreete math very much except that graph theory is cool
i took analysis as part of an advanced calculus class at third year it was the first 2 weeks of the class even at 30hrs per week of hw in it i still barely understand the crap.
on the bright side my university split it up into 3 classes so no one else has to suffer through it
i have 2, introduction to linear algebra for science and engineering and someone gave me linear algebra by Gilbert strange which is somehow alot worse then the first one
This question is related to one asked by somebody else here; however, unlike them, I do not want to show that a vector space over a field $F$ and of dimension $n$ is isomorphic to the $n$-th direct power $F^{n}$ of $F$. Instead, I want to generalize this to the following case:
If $B$ is a bas...
Suppose that $Z[x]$ is composed of all numbers of the form $c + dx$ where $c$ and d are integers and $x$ is an irrational positive solution to the polynomial $x^2 + b = 0$ where $b$ is an integer.
I seek to show that there exist any elements that are units aside from $1$ and $-1$. I know that...
