How to proceed after the below step. Also is there a simpler way to solve this integral?
how i evaluate this integral with integration by substitution $x=\frac{\pi}{2}-t$  $$\int _0^{\frac{\pi }{2}}\:\frac{\sqrt[3]{\sin^2\left(x\right)}}{\sqrt[3]{\sin^2\left(x\right)}+\sqrt[3]{\cos^2\left(x\right)}}dx$$
Are the limits $\lim\limits_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|\,$ and $\lim\limits_{n\to \infty }\sqrt[n]{|a_n|}\,$ equal? Let consider the power series $$ \sum_{k=1}^\infty a_kx^k. $$ We know that it converges when  $$\lim_{n\to \infty }\sqrt[n]{|a_n|}|x|&lt; 1\quad\text{or}\quad |x|&lt; \frac{1}{\lim_{n\to \infty }\sqrt[n]{|a_n|}}$$  But D'Alembert test tells us that it converges if $$|x|&lt; \frac{1}{\lim_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|},$$ so since the radius of convergence is unique, we should have that $$\lim_{n\to \infty }\sqrt[n]{|a_n|}=\lim_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|.$$  I tried to prove it, but this result looks strange to me. So if it doesn't work always, how can we have those to limit as radius of convergence? I guess the equality of those two limit should be true most of the time.  So under what conditions is it true? How do I disable the resizable property of a textarea? Linear mixed effect model interpretation with log transformed dependent variable I have a dataset in which the response variable is non normal, but on log transforming, it follows the normal distribution. I constructed a mixed effect model using lme4::lmer() as below (multiple measurements within each region), where A is a continuous variable and there are 7 regions  model&lt;-lmer(log(response)~A + (1|Region), data, REML=FALSE)   My output is   Linear mixed model fit by maximum likelihood . t-tests use Satterthwaite's method ['lmerModLmerTest'] Formula: logresponse ~ A + (1 | Region) Data: df[, -c(6)]   AIC      BIC   logLik deviance df.resid  151.8    159.8    -71.9    143.8       51   Scaled residuals:   Min       1Q   Median       3Q      Max  -2.24856 -0.63631  0.02206  0.59999  1.94895   Random effects: Groups   Name        Variance Std.Dev. Region   (Intercept) 0.8611   0.9280   Residual             0.5814   0.7625   Number of obs: 55, groups:  Region, 7  Fixed effects:               Estimate Std. Error        df t value Pr(&gt;|t|)   (Intercept) -1.112047   0.747373 19.651457  -1.488   0.1526    A           0.008585   0.004250 29.157266   2.020   0.0526 .  Correlation of Fixed Effects: (Intr) A -0.871   1) Would I infer that for A effects the response variable and the response increases by exp(0.008585) +/- exp(0.004250) (std error)? Would that be the right interpretation?  2) exp(Intercept) is not equal to the mean(log response). They are close but not equal. Why are they not equal and should I be concerned?   I know that this is a hard question to answer without knowing the dataset. So please suggest how I can improve the question. Book about classical mechanics I am looking for a book about "advanced" classical mechanics. By advanced I mean a book considering directly Lagrangian and Hamiltonian formulation, and also providing a firm basis in the geometrical consideration related to these to formalism (like tangent bundle, cotangent bundle, 1-form, 2-form, etc.).  I have , but I would like to go into more details about the [symplectic] geometrical and mathematical foundations of classical mechanics.  Additional note: A chapter about relativistic Hamiltonian dynamics would be a good thing. My ListView (of my ViewFilipper) fills the parent and buttons on the bottom get hidden River flowing down a slope Changing CSV layer attribute value from text to number in QGIS? I read in a CSV file for linking to a .shp file.   All the variables that are just numbers came in as text.   How do I change the text variables to numeric in QGIS? Should I submit bank statements when applying for a UK Visa? What do they say about me? Simple Proof by induction: $9$ divides $n^3 + (n+1)^3 + (n+2)^3$ I'm trying to prove using induction that $9$ divides $n^3 + (n+1)^3 + (n+2)^3$ whenever $n$ is a non-negative integer. So far, I have:   Base case: $P(1) = (1) + (8) + (27) = 36, 36$ can be divided by $9$ so the base case is valid Inductive step: let $P(n)$ be the statement $9$ divides $n^3 + (n+1)^3 + (n+2)^3$. Assume $P(k)$ is true, so $9$ divides $k^3 + (k+1)^3 + (k+2)^3$.   And this is where I'm stuck. I'm not sure how to demonstrate that $9$ divides $P(n)$ when $n = k+1$. If someone could step me in the right direction that would be awesome. Thanks! After the downfall of Voldemort, what happened to the Death Eaters?  They committed many crimes, including mass murder and torture. Nothing in the books made mention of their punishments. What happened? Layout selector missing The arrival of Minecraft Beta 1.5 brought weather conditions. However, it can at times be inconvenient when you're trying to screenshot something and show it off (as it reduces lighting and obstructs your view slightly). Is there anyway to turn weather off?  Alternatively, is there a way to speed up the duration of weather or "skip" it? Why pass parameters to CSS and JavaScript link files like src="../cnt.js?ver=4.0"? How to pass command output as multiple arguments to another command How to sanity check a date in Java I find it curious that the most obvious way to create Date objects in Java has been deprecated and appears to have been "substituted" with a not so obvious to use lenient calendar.  How do you check that a date, given as a combination of day, month, and year, is a valid date?  For instance, 2008-02-31 (as in yyyy-mm-dd) would be an invalid date. Most people say I wish I could, I wish you would.    Can we use I wish I can, I wish you will?    I'd like to know what the main differences between the usage of can/will and could/would are when wishing.  Is it related to the subjunctive? Because one says “I wish I were”, not “I wish I am”, so are would and could the subjunctive forms of can and will?  Or is it something else entirely? 2D inhomogeneous biharmonic equation How to stop constantly blinking Wifi LED? How can I make iconv replace the input file with the converted output? I need to prove that $\mathbb{R}\backslash\mathbb{Q} \sim \mathbb{R} $  Using Cantor-Bernstein, need to show an injection from $\mathbb{R}\backslash\mathbb{Q}$ to $\mathbb{R}$ and from $\mathbb{R}$ to $\mathbb{R}\backslash\mathbb{Q}$.    $\mathbb{R}\backslash\mathbb{Q}$ is a subset of $\mathbb{R}$ so only need to show injection from $\mathbb{R}$ to $\mathbb{R}\backslash\mathbb{Q}$ to complete the proof.    Possible injection: $f:\mathbb{R}\to \mathbb{R}\backslash\mathbb{Q}$ defined as $f(x) = \pi x$ if $x$ is not a multiple of $\pi$; otherwise $f(x) = \sqrt{2} x$.    Not sure if $f$ actually is an injection...