Determine the domain of definition and the domain
Determine the domain of definition and the domain of continuity of $\displaystyle f(x)=\sum_{n=1}^{\infty}\left(x+\frac1n\right)^n$.
Explanation of Method
The Root Test gives $\sqrt[n]{|u_n(x)|}=|x+1/n|\to|x|$, so the series converges for $|x|<1$ and diverges for $|x|>1$; the boundary cases $x=\pm1$ need separate checking since $(1+1/n)^n\to e$ and its sibling at $x=-1$ both fail to tend to $0$. Uniform convergence on compact subsets of $(-1,1)$ then gives continuity there via the M-test.
- Apply the Root Test to locate the interval of convergence.
- Check the boundary points $x=\pm1$ via the Divergence Test.
- Establish uniform convergence on compact subsets and conclude continuity.
Step 1 · Root Test
$\sqrt[n]{|u_n(x)|}=\left|x+\dfrac1n\right|\to|x|$ as $n\to\infty$. So the series converges for $|x|<1$ and diverges for $|x|>1$.
Step 2 · Boundary points
At $x=1$: $\left(1+\dfrac1n\right)^n\to e\ne0$, so the series diverges (Divergence Test). At $x=-1$: $\left(-1+\dfrac1n\right)^n=(-1)^n\left(1-\dfrac1n\right)^n$, and since $\left(1-\dfrac1n\right)^n\to e^{-1}$, the terms oscillate with magnitude approaching $e^{-1}\ne0$ — they do not tend to $0$, so the series diverges here as well.
Step 3 · Domain, uniform convergence, and continuity
The domain of definition is $(-1,1)$. For any compact $[a,b]\subset(-1,1)$, let $r=\max(|a|,|b|)<1$; for $n$ large enough, $\left|x+\dfrac1n\right|\le r+\dfrac1n<\rho$ for some fixed $\rho<1$, so $|u_n(x)|\le\rho^n$ uniformly on $[a,b]$. Since $\sum\rho^n$ converges, the Weierstrass M-test gives uniform convergence on $[a,b]$, and since each term is continuous, $f$ is continuous on $[a,b]$ — hence on all of $(-1,1)$.
Summary Table
| Property | Value |
|---|---|
| Root test limit | $|x|$ |
| At $x=1$ | Diverges (terms $\to e$) |
| At $x=-1$ | Diverges (terms oscillate, magnitude $\to1/e$) |
| Domain of definition | $(-1,1)$ |
| Domain of continuity | $(-1,1)$ |
Additional Notes
- This problem's domain equals its continuity domain exactly, in contrast with Problem 206 where the domain $[0,\infty)$ strictly exceeds the continuity domain $(0,\infty)$ — here uniform convergence on every compact subset of the open interval $(-1,1)$ suffices to make the two sets coincide.
- The failure at $x=1$ recycles the exact limit $(1+1/n)^n\to e$ used throughout Problem 183's analysis of $(1+x/n)^n\to e^x$; here that same limit shows up as the reason the series can never converge at the right endpoint.
- Since the series cannot be extended continuously to $x=\pm1$ (the terms simply fail to vanish there), this is a genuine "hard" boundary — unlike the removable discontinuity of Problem 206, there is no way to patch the definition at $x=\pm1$ to recover convergence.
Result
Domain of definition and domain of continuity coincide: