Cálculo Ejemplos

$\sum_{k = 1}^{3} \frac{1}{k^{2} + 4}$

Paso 1

Expande la serie para cada valor de $k$ .

$\frac{1}{1^{2} + 4} + \frac{1}{2^{2} + 4} + \frac{1}{3^{2} + 4}$

Paso 2

Simplifica.

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Paso 2.1

Obtén el denominador común

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Paso 2.1.1

Multiplica $\frac{1}{1^{2} + 4}$ por $\frac{(2^{2} + 4) (3^{2} + 4)}{(2^{2} + 4) (3^{2} + 4)}$ .

$\frac{1}{1^{2} + 4} \cdot \frac{(2^{2} + 4) (3^{2} + 4)}{(2^{2} + 4) (3^{2} + 4)} + \frac{1}{2^{2} + 4} + \frac{1}{3^{2} + 4}$

Paso 2.1.2

Multiplica $\frac{1}{1^{2} + 4}$ por $\frac{(2^{2} + 4) (3^{2} + 4)}{(2^{2} + 4) (3^{2} + 4)}$ .

$\frac{(2^{2} + 4) (3^{2} + 4)}{(1^{2} + 4) ((2^{2} + 4) (3^{2} + 4))} + \frac{1}{2^{2} + 4} + \frac{1}{3^{2} + 4}$

Paso 2.1.3

Multiplica $\frac{1}{2^{2} + 4}$ por $\frac{(1^{2} + 4) (3^{2} + 4)}{(1^{2} + 4) (3^{2} + 4)}$ .

$\frac{(2^{2} + 4) (3^{2} + 4)}{(1^{2} + 4) ((2^{2} + 4) (3^{2} + 4))} + \frac{1}{2^{2} + 4} \cdot \frac{(1^{2} + 4) (3^{2} + 4)}{(1^{2} + 4) (3^{2} + 4)} + \frac{1}{3^{2} + 4}$

Paso 2.1.4

Multiplica $\frac{1}{2^{2} + 4}$ por $\frac{(1^{2} + 4) (3^{2} + 4)}{(1^{2} + 4) (3^{2} + 4)}$ .

$\frac{(2^{2} + 4) (3^{2} + 4)}{(1^{2} + 4) ((2^{2} + 4) (3^{2} + 4))} + \frac{(1^{2} + 4) (3^{2} + 4)}{(2^{2} + 4) ((1^{2} + 4) (3^{2} + 4))} + \frac{1}{3^{2} + 4}$

Paso 2.1.5

Multiplica $\frac{1}{3^{2} + 4}$ por $\frac{(1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4)}$ .

$\frac{(2^{2} + 4) (3^{2} + 4)}{(1^{2} + 4) ((2^{2} + 4) (3^{2} + 4))} + \frac{(1^{2} + 4) (3^{2} + 4)}{(2^{2} + 4) ((1^{2} + 4) (3^{2} + 4))} + \frac{1}{3^{2} + 4} \cdot \frac{(1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4)}$

Paso 2.1.6

Multiplica $\frac{1}{3^{2} + 4}$ por $\frac{(1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4)}$ .

$\frac{(2^{2} + 4) (3^{2} + 4)}{(1^{2} + 4) ((2^{2} + 4) (3^{2} + 4))} + \frac{(1^{2} + 4) (3^{2} + 4)}{(2^{2} + 4) ((1^{2} + 4) (3^{2} + 4))} + \frac{(1^{2} + 4) (2^{2} + 4)}{(3^{2} + 4) ((1^{2} + 4) (2^{2} + 4))}$

Paso 2.1.7

Reordena los factores de $(1^{2} + 4) ((2^{2} + 4) (3^{2} + 4))$ .

$\frac{(2^{2} + 4) (3^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)} + \frac{(1^{2} + 4) (3^{2} + 4)}{(2^{2} + 4) ((1^{2} + 4) (3^{2} + 4))} + \frac{(1^{2} + 4) (2^{2} + 4)}{(3^{2} + 4) ((1^{2} + 4) (2^{2} + 4))}$

Paso 2.1.8

Reordena los factores de $(2^{2} + 4) ((1^{2} + 4) (3^{2} + 4))$ .

$\frac{(2^{2} + 4) (3^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)} + \frac{(1^{2} + 4) (3^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)} + \frac{(1^{2} + 4) (2^{2} + 4)}{(3^{2} + 4) ((1^{2} + 4) (2^{2} + 4))}$

Paso 2.1.9

Reordena los factores de $(3^{2} + 4) ((1^{2} + 4) (2^{2} + 4))$ .

$\frac{(2^{2} + 4) (3^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)} + \frac{(1^{2} + 4) (3^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)} + \frac{(1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Paso 2.2

Combina los numeradores sobre el denominador común.

$\frac{(2^{2} + 4) (3^{2} + 4) + (1^{2} + 4) (3^{2} + 4) + (1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Paso 2.3

Simplifica cada término.

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Paso 2.3.1

Eleva $2$ a la potencia de $2$ .

$\frac{(4 + 4) (3^{2} + 4) + (1^{2} + 4) (3^{2} + 4) + (1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Paso 2.3.2

Suma $4$ y $4$ .

$\frac{8 (3^{2} + 4) + (1^{2} + 4) (3^{2} + 4) + (1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Paso 2.3.3

Eleva $3$ a la potencia de $2$ .

$\frac{8 (9 + 4) + (1^{2} + 4) (3^{2} + 4) + (1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Paso 2.3.4

Suma $9$ y $4$ .

$\frac{8 \cdot 13 + (1^{2} + 4) (3^{2} + 4) + (1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Paso 2.3.5

Multiplica $8$ por $13$ .

$\frac{104 + (1^{2} + 4) (3^{2} + 4) + (1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Paso 2.3.6

Uno elevado a cualquier potencia es uno.

$\frac{104 + (1 + 4) (3^{2} + 4) + (1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Paso 2.3.7

Suma $1$ y $4$ .

$\frac{104 + 5 (3^{2} + 4) + (1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Paso 2.3.8

Eleva $3$ a la potencia de $2$ .

$\frac{104 + 5 (9 + 4) + (1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Paso 2.3.9

Suma $9$ y $4$ .

$\frac{104 + 5 \cdot 13 + (1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Paso 2.3.10

Multiplica $5$ por $13$ .

$\frac{104 + 65 + (1^{2} + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Paso 2.3.11

Uno elevado a cualquier potencia es uno.

$\frac{104 + 65 + (1 + 4) (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Paso 2.3.12

Suma $1$ y $4$ .

$\frac{104 + 65 + 5 (2^{2} + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Paso 2.3.13

Eleva $2$ a la potencia de $2$ .

$\frac{104 + 65 + 5 (4 + 4)}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Paso 2.3.14

Suma $4$ y $4$ .

$\frac{104 + 65 + 5 \cdot 8}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Paso 2.3.15

Multiplica $5$ por $8$ .

$\frac{104 + 65 + 40}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Paso 2.4

Suma $104$ y $65$ .

$\frac{169 + 40}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Paso 2.5

Suma $169$ y $40$ .

$\frac{209}{(1^{2} + 4) (2^{2} + 4) (3^{2} + 4)}$

Paso 2.6

Simplifica el denominador.

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Paso 2.6.1

Uno elevado a cualquier potencia es uno.

$\frac{209}{(1 + 4) (2^{2} + 4) (3^{2} + 4)}$

Paso 2.6.2

Suma $1$ y $4$ .

$\frac{209}{5 (2^{2} + 4) (3^{2} + 4)}$

Paso 2.6.3

Eleva $2$ a la potencia de $2$ .

$\frac{209}{5 (4 + 4) (3^{2} + 4)}$

Paso 2.6.4

Suma $4$ y $4$ .

$\frac{209}{5 \cdot 8 (3^{2} + 4)}$

Paso 2.6.5

Multiplica $5$ por $8$ .

$\frac{209}{40 (3^{2} + 4)}$

Paso 2.6.6

Eleva $3$ a la potencia de $2$ .

$\frac{209}{40 (9 + 4)}$

Paso 2.6.7

Suma $9$ y $4$ .

$\frac{209}{40 \cdot 13}$

Paso 2.7

Multiplica $40$ por $13$ .

$\frac{209}{520}$

Paso 3

El resultado puede mostrarse de distintas formas.

Forma exacta:

$\frac{209}{520}$

Forma decimal:

$0.40192307 \dots$