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For a function $(1+x)^p$, when we write the power series and use the ratio test to find out the interval of convergence, we get $|x|<1$. But take a positive value of $p$, 3 for example. The function becomes $(1+x)^3$ and surely the power series reduces to $1+3x+3x^2+x^3$, and This gives a finite value for any value of $x$ not just for $|x|<1$, so what is it that I'm missing ?

Kooranifar
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R Premjith
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  • Hi! Welcome to Math Stack Exchange. Please edit your post, using math formattation because this is unreadable. Thanks. – Matteo Nov 25 '21 at 20:30
  • The convergence of the power series is from the center to the nearest singularity. $(1+x)^p=\exp(p\ln(1+x))$. The $\ln(1+x)$ has a singularity at $x=-1$ and most of the time multiplying by $p$ and taking $\exp$ doesn't remove it. But as you saw, for $p\in\mathbb{N}$ it does. The most clear reason of how it gets removed I think it is the computation that you did of showing that the function coincides with a polynomial. – plop Nov 25 '21 at 20:40
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    You should write the derivation that you say gives $|x|<1$. Then we can tell you what step changes if $p$ is a natural number. – Milten Nov 25 '21 at 21:39
  • Your caveat applies for p a positive integer, where we have an ordinary polynomial. Fractional or negative p are different stories. – herb steinberg Nov 25 '21 at 23:16
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    If $p$ is a positive integer the coefficients in the power series at hand will be zero eventually, hence the ratio test will be quite problematic..! – Jakob Streipel Nov 26 '21 at 00:52

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