Series
Σaₙ converges when its partial sums converge. Absolute convergence means Σ|aₙ|<∞.
Find a concept
REFERENCE SHEET
Series & uniform convergence
A pointwise limit answers one input at a time; a uniform limit controls all inputs at once.
Before this Completeness & sequences, Integration & accumulation
fₙ→f uniformly ⇔ supₓ |fₙ(x)−f(x)|→0
Definitions
Pointwise convergence
For each fixed x, fₙ(x) tends to f(x); how late n must be may depend on x.
Uniform convergence
For every ε>0 a single N makes |fₙ(x)−f(x)|<ε for all x in the domain and all n≥N.
Key theorems
Uniform limit theorem
A uniform limit of continuous real- or complex-valued functions is continuous.
Weierstrass M-test
If |fₙ(x)|≤Mₙ for all x and ΣMₙ<∞, then Σfₙ converges absolutely at every x and uniformly on the domain.
Integration under uniform limits
If Riemann-integrable fₙ on [a,b] converge uniformly to f, then f is Riemann integrable and ∫fₙ→∫f.
Notable applications
Approximation
Uniform error bounds justify replacing a function throughout an interval.
Power series
Inside a power series radius of convergence, compact subintervals admit strong uniform control.
Work it out
Try before revealing01 / Compare
Find the pointwise limit of xⁿ on [0,1]. Is convergence uniform?
Hint
Separate x<1 from x=1.
Solution
The limit is 0 for x<1 and 1 at x=1. It is discontinuous, while each xⁿ is continuous, so convergence is not uniform.
02 / Prove
Show Σₙ₌₁∞ xⁿ/n² converges uniformly on [−1,1].
Hint
Bound each term independently of x.
Solution
|xⁿ/n²|≤1/n² and Σ1/n² converges. The M-test gives uniform convergence.
03 / Counterexample
For fₙ(x)=xⁿ on [0,1], find the pointwise limit and prove convergence is not uniform.
Hint
What happens at 1, and what does a uniform limit preserve?
Solution
The limit is 0 for x<1 and 1 at x=1. It is discontinuous, whereas each fₙ is continuous; a uniform limit would be continuous. Also supₓ∈[0,1]|fₙ(x)−f(x)|=1 for every n.
Connect the ideas
Build on this
These sheets use this topic as a prerequisite.
Go deeper
Further reading and full-length treatments: