Intereting Posts

How to prove the existence of $b$ in $Q$ such that $a<b^2<c$ in $Q$?
Proving the number of even and odd permutations of a subgroup $H<S_{n}$ are equal, provided $H$ is not contained in $A_{n}$
is there a cardinality between the rational and the irrationals?
General question about mathematical thinking
What is the mathematical significance of Penrose tiles?
Percentage of Composite Odd Numbers Divisible by 3
Is there an analytic solution for the equation $\log_{2}{x}+\log_{3}{x}+\log_{4}{x}=1$?
How to find the number of distinct combinations of a non distinct set of elements?
Are there vector bundles that are not locally trivial?
Different arrows in set theory: $\rightarrow$ and $\mapsto$
finite dimensional range implies compact operator
Striking applications of integration by parts
$C$ is complete w.r.t. which norm(s)
Proof that $\dim(U_1+U_2) = \dim U_1 + \dim U_2 – \dim(U_1\cap U_2)$
Proof of Dedekind's Theorem on the Galois Groups of rational polynomials

I’m looking for a technique for creating alternating negatives and positives in a series. Specifically: when n=1, the answer is +, n=2 is +, n=3 is -, n=4 is -… etc.

I have every other part of the series written but I can’t figure out that last piece… here’s what I have now:

$$\sum_1^\infty 2^{n-1}(1^n+(-1)^n)/(3^{n-1}n!)*x^n$$

- How to calculate $2^{\sqrt{2}}$ by hand efficiently?
- A closed form of the series $\sum_{n=1}^{\infty} \frac{H_n^2-(\gamma + \ln n)^2}{n}$
- Reference request: an analytical proof the Hilbert space filling curve is nowhere differentiable
- Integrating $\int\frac{x^3}{\sqrt{9-x^2}}dx$ via trig substitution
- Rectifiability of a curve
- Solve $\lim_{x\to 0} \frac{\sin x-x}{x^3}$

Technically, every other term is 0 so there doesn’t really need to be two negatives in a row, it just has to sync up where I need them–I’m just guessing that I’d need it to work that way. Thanks for your assistance!

- Method of Exhaustion applied to Parabolic Segment in Apostol's Calculus
- How to Evaluate $ \int \! \frac{dx}{1+2\cos x} $ ?
- What are some easy to understand applications of Banach Contraction Principle?
- How to calclulate a derivate of a hypergeometric function w.r.t. one of its parameters?
- How to calculate this expectation where the random variable is restricted on a sphere?
- induction proof: $\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$
- How to prove that exponential grows faster than polynomial?
- Help solving a limit in two parts $\lim_{t\to 0}\left(\frac{1}{t\sqrt{1+t}}-\frac{1}{t}\right)$
- Solutions of autonomous ODEs are monotonic
- $\{a_n\}$ sequence $a_1=\sqrt{6}$ for $n \geq 1$ and $a_{n+1}=\sqrt{6+a_n}$ show it that convergence and also find $\lim_{x \to \infty} \{a_n\}$

Here’s a little trick which you might appreciate, or which you might find to obfuscate the matter. The triangular numbers $1,3,6,10,15,\dots$, given by the formula $T_n=n(n+1)/2$, have the property that $T_{4k+1}$ and $T_{4k+2}$ are odd and that $T_{4k+3}$ and $T_{4k+4}$ are even. So the expression $-(-1)^{T_n}=(-1)^{T_n+1} = (-1)^{(n^2+n+2)/2}$ alternates in sign in the way that you are looking for. In other words, if your original series is $\sum_{n=1}^\infty a_n$, where all $a_n$ are positive, then the modified alternating series you want is

$$

\sum_{n=1}^\infty (-1)^{(n^2+n+2)/2}a_n.

$$

$$S=\sum^\infty_{n=1}(-1)^{n+1}f(2n-1)+\sum^\infty_{n=1}(-1)^{n+1}f(2n)=f(1)+f(2)-f(3)-f(4)+…$$

$$ \sqrt 2 \; \sin \left( \frac{(2n-1)\pi}{4} \right) $$

Try: $$a_n = (-1)^{\frac{\left(2n+1+(-1)^{n+1}\right)}{4}+1}$$

Then, $$\langle a_n\rangle = 1, 1, -1, -1, 1, 1, \ldots$$

Derivation:

*“Any sufficiently advanced technology is indistinguishable from magic.” – Clark’s Third Law*

I don’t really know how to describe how I got that… `;)`

I must confess that I do not see the relation between your interesting questions and the expression you give. In practice,$$\sum_1^\infty 2^{n-1}(1^n+(-1)^n)/(3^{n-1}n!)*x^n=3 \left(\cosh \left(\frac{2 x}{3}\right)-1\right)$$ just involves even powers of $x$.

- Optimal yarn balls
- Find $ \int \frac {\tan 2x} {\sqrt {\cos^6x +\sin^6x}} dx $
- If $\int_{0}^{\infty} f(x) \, dx $ converges, will $\int_{0}^{\infty}e^{-sx} f(x) \, dx$ always converge uniformly on $[0, \infty)$?
- Distinct digits in a combination of 6 digits
- Where can I learn more about commutative hyperoperations?
- Probability of 3 Heads in 10 Coin Flips
- How is the general solution for algebraic equations of degree five formulated?
- Show that $N_n \mid N_m$ if and only if $n \mid m$
- Secretary problem – why is the optimal solution optimal?
- Using calculus of residues to evaluate a trig integral
- Expectation of an exponential function
- Is there a name for this strange solution to a quadratic equation involving a square root?
- fundamental group of $U(n)$
- How to evaluate the trigonometric integral $\int \frac{1}{\cos x+\tan x }dx$
- Has error correction been “solved”?