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The power series for log(1+x) on (-1,1], including the Abel endpoint
Statement
For , The series converges at to and diverges at .
Facts & Assumptions
Given: A real with .
For , (For , , and for the series diverges).
A real power series may be integrated term by term inside its radius of convergence (Inside its radius a real power series may be integrated term by term on every closed subinterval).
The alternating harmonic series converges, Abel's limit theorem identifies the limit at of a convergent power series with its sum, and the harmonic series diverges (The alternating series test: if is nonincreasing with then converges, the sum lies between any two consecutive partial sums, and the error after terms is at most , Abel's limit theorem: if a real series converges to , then its power series tends to as , For rational , converges iff ).
Proof
Integrating the series of [L2] from to gives .
By [L1], the integral of from to is , so the displayed series formula holds for .
At the series is alternating harmonic and converges by [L4]; Abel's theorem and step 2.1 identify its sum with .
At every term is , so the series is the negative harmonic series and diverges.
Depends on
- The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t
- For $|r| < 1$, $\sum_{k \ge 0} r^k = 1/(1-r)$, and for $|r| \ge 1$ the series diverges
- Inside its radius a real power series may be integrated term by term on every closed subinterval
- The alternating series test: if $(b_k)$ is nonincreasing with $b_k \to 0$ then $\sum_{k} (-1)^{k} b_k$ converges, the sum lies between any two consecutive partial sums, and the error after $n$ terms is at most $b_n$
- Abel's limit theorem: if a real series converges to $s$, then its power series tends to $s$ as $x\uparrow1$
- For rational $p > 0$, $\sum 1/k^p$ converges iff $p > 1$
Used by
- The log(1+x) power series diverges at x=-1 Counterexample
- Abel convergence of the alternating harmonic power series along a nonradial Stolz approach Example
- log 2 from four characterisations: hyperbolic area, the alternating harmonic series, Landau iterates and exp⁻¹(2) Example
- The alternating harmonic power series tends to log 2 at the boundary point 1 Example
- The alternating harmonic series sums to log 2 Example
- Stirling's factorial asymptotic holds up to a positive constant Lemma
- The Mercator series, its value at 1 and the product law determine log on all positive reals, while the series alone is only local Lemma
- Five characterisations of the natural logarithm are equivalent: inverse exponential, integral, continued Mercator series, Landau root limit and the normalised functional equation Theorem
- Mertens' third theorem for primes Theorem
- Standard Maclaurin expansions Theorem
- Under square summability, the signed product of (1+pₙ) converges iff the series of pₙ converges Theorem
Dependency tree · two levels
45 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)
- MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)