Calculus assignment: the function ln(1 + x)/x and its associated sequence
A complete take-home assignment: bound ln(1 + x) by integration, study the function f(x) = ln(1 + x)/x, then deduce the convergence speed of the sequence u(n+1) = ln(1 + u(n)).
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Table of contents
- 01
Part I — Bounding ln(1 + x) and studying f
- Bounding ln(1 + x) by integration(not finished)
- The auxiliary function g, and what it brings(not finished)
- Bounding 1/(1 + t)Open answer
- From 1/(1 + t) to ln(1 + x)Open answer
- Bounding the derivative of the auxiliary function gOpen answer
- The resulting bound on gMCQ
- The variations of fOpen answer
- The limit of f at +∞Short answer
- The limit that yields differentiability at 0Open answer
- The tangent to the curve of f at 0Short answer
- The shape of the curve of fOpen answer
- 02
Part II — The sequence defined by u(n+1) = ln(1 + u(n))