(f_{n})_{n∈} :→

f_{n}(x) = \frac{nx}{n^{2} + x^{2}} .

(f_{n})_{n∈} : [0, +∞) →

f_{n}(x) = \begin{cases}\frac{2n}{n + x} & x ≥ n \\ 1 & 0 ≤ x ≤ n. \end{cases}

(f_{n})_{n∈} :→

f_{n}(x) = \frac{nx}{1 + n^{2}x^{2}} .

(f_{n})_{n∈} : [0, +∞) →

f_{n}(x) = ln (1 + \frac{x}{n^{2}} ) , x ≥ 0.

(f_{n})_{n∈} : [0, 1] →

f_{n}(x) = \begin{cases}nx & 0 ≤ x ≤ 1/n \\ −\frac{nx}{n − 1} + \frac{n}{n − 1} & 1/n ≤ x ≤ 1. \end{cases}

(f_{n})_{n∈} :→ f_{n}(x) = arctan(nx).

(f_{n})_{n∈} :→

f_{n}(x) = \sqrt{x^{2} + \frac{1}{n}}.

(f_{n})_{n∈} : [0, 1] → f_{n}(x) = x(1−x)^{n} ln n, x ∈ [0, 1].

(f_{n})_{n∈} :→

f_{n}(x) = \frac{x + n}{x^{2} + n}.

lim_{n→∞} ∫_{0}^{1} \frac{x^{2}}{x + n} sin nxdx.

(g_{n})_{n∈} :→

g_{n}(x) = \frac{1}{\sqrt{x^{2} + n}},

f_{n}(x) = ∫_{0}^{x} g_{n}(t)dt.

(f_{n})_{n∈} : [0, 1] →

f_{n}(x) = n^{α}x(1−x^{2})^{n}, α > 0,

lim_{n→+∞} ∫_{0}^{1} f_{n}(x)dx.

(f_{n})_{n∈} : [0, π] →

f_{n}(x) = (\frac{1}{n} + sin^{2} x)^{n} ,

lim_{n→+∞} ∫_{0}^{π} f_{n}(x)dx.

` e

lim_{n→∞} ∫_{I} f_{n}(x)dx = ∫_{I} lim_{n→∞} f_{n}(x)dx.

\begin{matrix}g_{n}(x) = \sqrt{n}χ_{[1/(2n),1/n]}(x) & I = [0, 1] \\ h_{n}(x) = (x − n)χ_{[n,n+1]}(x) + (n + 2 − x)χ_{[n+1,n+2]} & I = \\ k_{n}(x) = χ_{[n.n+1]}(x) & I = \end{matrix}

(f_{n})_{n∈} : [0, 1] →

f_{1}(x) = \begin{cases}x & x ∈ [0, 1/2] \\ 1 − x & x ∈ [1/2, 1], \end{cases}

f_{n+1}(x) = \begin{cases}\frac{1}{2}f_{n}(2x) & x ∈ [0, 1/2] \\ \frac{1}{2}f_{n}(2x − 1) & x ∈ [1/2, 1]. \end{cases}

f_{1}(x) = \begin{cases}\frac{3}{2}x & x ∈ [0, 1/3] \\ \frac{1}{2} & x ∈ [1/3, 2/3] \\ \frac{3}{2}x − \frac{1}{2} & x ∈ [2/3, 1], \end{cases}

f_{n+1}(x) = \begin{cases}\frac{1}{2}f_{n}(3x) & x ∈ [0, 1/3] \\ \frac{1}{2} & x ∈ [1/3, 2/3] \\ \frac{1}{2}f_{n}(3x − 2) + \frac{1}{2} & x ∈ [2/3, 1]. \end{cases}

∑_{n=1}^{∞} \sqrt{n} log (1 + \frac{x}{n^{2}} ) , x ≥ 0.

∑_{n=1}^{∞}(−1)^{n} \frac{1}{n + x}, x ≥ 0.

∑_{n=1}^{∞}(−1)^{n} \frac{x^{n}}{n} , x ∈ [0, 1].

∑_{n=1}^{∞} n (\frac{x + 1}{x})^{n} .

∑_{n=0}^{∞} \frac{1}{n + 1} (\frac{x}{2})^{n+1} .

∑_{n=1}^{∞} \frac{n + x}{n^{3}x^{2} + 1}, x ∈ .

∑_{n=1}^{∞} \frac{x}{n}e^{−nx}, x ∈ .

p ∈

∑_{n=1}^{∞} \frac{x}{n^{p}(1 + nx^{2})}, x ∈ .

∑_{n=1}^{∞} \frac{1}{n^{x}} , x ≥ 0.

∑_{n=1}^{∞}(−1)^{n} \frac{1}{n^{x}} , x ≥ 0.

F (x) = ∑_{n=1}^{∞} f_{n}(x)

(f_{n})_{n∈} : [0, 1] →

è à à à F x = 0

f(x) = log (\frac{2 + x}{1 + x^{2}} ) .

x = 0 f(x) = (1+x)^{α}.

2π [0, 2π]

f(x) = x.

2π [0, 2π]

f(x) = x^{2}.

2π [0, π]

f(x) = x^{2}−πx.