Determine the radius of convergence and test convergence
Determine the radius of convergence and test convergence at the endpoints for $\displaystyle\sum_{n=1}^{\infty}\left(\frac{n+1}n\right)^{n^2}x^n$.
Explanation of Method
The Root Test is the natural tool since the coefficient is already a clean $n$-th power raised to the $n$-th power: $\sqrt[n]{a_n}=(1+1/n)^n\to e$, giving radius $1/e$. A refined asymptotic (via the Taylor expansion of $\ln(1+1/n)$) shows the terms at either endpoint tend to a nonzero constant, so both endpoints diverge.
- Apply the Root Test using $\sqrt[n]{a_n}=(1+1/n)^n\to e$.
- Refine the asymptotic of $a_n$ via the Taylor series of $\ln(1+1/n)$.
- Show the terms at $x=\pm1/e$ tend to a nonzero constant, forcing divergence.
Step 1 · Radius of convergence via the Root Test
With $a_n=(1+1/n)^{n^2}$: $$ \sqrt[n]{a_n} = \left(1+\frac1n\right)^n \longrightarrow e. $$ So $R=1/e$.
Step 2 · Refine the asymptotic of $a_n$
Using $\ln(1+t)=t-\dfrac{t^2}2+O(t^3)$ with $t=1/n$: $$ n^2\ln\left(1+\frac1n\right) = n^2\left(\frac1n-\frac1{2n^2}+O\!\left(\frac1{n^3}\right)\right) = n-\frac12+O\!\left(\frac1n\right). $$ So $$ a_n = e^{n-1/2+O(1/n)} = \frac{e^n}{\sqrt e}\bigl(1+O(1/n)\bigr). $$
Step 3 · Endpoints $x=\pm1/e$
At $x=1/e$: $a_n\cdot\dfrac1{e^n} = \dfrac1{\sqrt e}(1+O(1/n)) \to \dfrac1{\sqrt e} \ne0$. At $x=-1/e$: $a_n\cdot\dfrac{(-1)^n}{e^n}\to\dfrac{(-1)^n}{\sqrt e}$, which oscillates but never approaches $0$ in magnitude. In both cases the terms fail to tend to $0$, so by the Divergence Test, the series diverges at both endpoints.
Summary Table
| Property | Value |
|---|---|
| Radius of convergence | $R=1/e$ |
| Term size at either endpoint | $\to1/\sqrt e\ne0$ |
| Interval of convergence | $(-1/e,\,1/e)$ |
Additional Notes
- This is the sharpest use of the limit $\lim(1+1/n)^n=e$ in this batch, applied via the Root Test rather than the Ratio Test — a natural choice whenever the coefficient is already expressed as an $n$-th power.
- The refined asymptotic $a_n\sim e^n/\sqrt e$ shows that the naive first-order approximation $\sqrt[n]{a_n}\to e$ (used only to locate the radius) is not precise enough to resolve the boundary behavior — a second-order Taylor term ($-1/(2n)$ inside the exponent) is exactly what's needed to pin down the endpoint constant $1/\sqrt e$.
- This coefficient growth rate, exponential in $n$ with a sub-exponential correction, sits between the polynomial-growth series of Problems 223–232 and the factorial/double-exponential coefficients of Problems 224, 235, and 240 — a useful reference point for how quickly a radius shrinks as coefficients grow.