Cálculo Ejemplos

$\sum_{j = 3}^{5} \frac{1}{j^{2} - 3}$

Paso 1

Expande la serie para cada valor de $j$ .

$\frac{1}{3^{2} - 3} + \frac{1}{4^{2} - 3} + \frac{1}{5^{2} - 3}$

Paso 2

Simplifica.

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Paso 2.1

Obtén el denominador común

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Paso 2.1.1

Multiplica $\frac{1}{3^{2} - 3}$ por $\frac{(4^{2} - 3) (5^{2} - 3)}{(4^{2} - 3) (5^{2} - 3)}$ .

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

Paso 2.1.2

Multiplica $\frac{1}{3^{2} - 3}$ por $\frac{(4^{2} - 3) (5^{2} - 3)}{(4^{2} - 3) (5^{2} - 3)}$ .

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

Paso 2.1.3

Multiplica $\frac{1}{4^{2} - 3}$ por $\frac{(3^{2} - 3) (5^{2} - 3)}{(3^{2} - 3) (5^{2} - 3)}$ .

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

Paso 2.1.4

Multiplica $\frac{1}{4^{2} - 3}$ por $\frac{(3^{2} - 3) (5^{2} - 3)}{(3^{2} - 3) (5^{2} - 3)}$ .

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

Paso 2.1.5

Multiplica $\frac{1}{5^{2} - 3}$ por $\frac{(3^{2} - 3) (4^{2} - 3)}{(3^{2} - 3) (4^{2} - 3)}$ .

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

Paso 2.1.6

Multiplica $\frac{1}{5^{2} - 3}$ por $\frac{(3^{2} - 3) (4^{2} - 3)}{(3^{2} - 3) (4^{2} - 3)}$ .

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

Paso 2.1.7

Reordena los factores de $(3^{2} - 3) ((4^{2} - 3) (5^{2} - 3))$ .

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

Paso 2.1.8

Reordena los factores de $(4^{2} - 3) ((3^{2} - 3) (5^{2} - 3))$ .

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

Paso 2.1.9

Reordena los factores de $(5^{2} - 3) ((3^{2} - 3) (4^{2} - 3))$ .

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

Paso 2.2

Combina los numeradores sobre el denominador común.

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

Paso 2.3

Simplifica cada término.

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Paso 2.3.1

Eleva $4$ a la potencia de $2$ .

$\frac{(16 - 3) (5^{2} - 3) + (3^{2} - 3) (5^{2} - 3) + (3^{2} - 3) (4^{2} - 3)}{(3^{2} - 3) (4^{2} - 3) (5^{2} - 3)}$

Paso 2.3.2

Resta $3$ de $16$ .

$\frac{13 (5^{2} - 3) + (3^{2} - 3) (5^{2} - 3) + (3^{2} - 3) (4^{2} - 3)}{(3^{2} - 3) (4^{2} - 3) (5^{2} - 3)}$

Paso 2.3.3

Eleva $5$ a la potencia de $2$ .

$\frac{13 (25 - 3) + (3^{2} - 3) (5^{2} - 3) + (3^{2} - 3) (4^{2} - 3)}{(3^{2} - 3) (4^{2} - 3) (5^{2} - 3)}$

Paso 2.3.4

Resta $3$ de $25$ .

$\frac{13 \cdot 22 + (3^{2} - 3) (5^{2} - 3) + (3^{2} - 3) (4^{2} - 3)}{(3^{2} - 3) (4^{2} - 3) (5^{2} - 3)}$

Paso 2.3.5

Multiplica $13$ por $22$ .

$\frac{286 + (3^{2} - 3) (5^{2} - 3) + (3^{2} - 3) (4^{2} - 3)}{(3^{2} - 3) (4^{2} - 3) (5^{2} - 3)}$

Paso 2.3.6

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

$\frac{286 + (9 - 3) (5^{2} - 3) + (3^{2} - 3) (4^{2} - 3)}{(3^{2} - 3) (4^{2} - 3) (5^{2} - 3)}$

Paso 2.3.7

Resta $3$ de $9$ .

$\frac{286 + 6 (5^{2} - 3) + (3^{2} - 3) (4^{2} - 3)}{(3^{2} - 3) (4^{2} - 3) (5^{2} - 3)}$

Paso 2.3.8

Eleva $5$ a la potencia de $2$ .

$\frac{286 + 6 (25 - 3) + (3^{2} - 3) (4^{2} - 3)}{(3^{2} - 3) (4^{2} - 3) (5^{2} - 3)}$

Paso 2.3.9

Resta $3$ de $25$ .

$\frac{286 + 6 \cdot 22 + (3^{2} - 3) (4^{2} - 3)}{(3^{2} - 3) (4^{2} - 3) (5^{2} - 3)}$

Paso 2.3.10

Multiplica $6$ por $22$ .

$\frac{286 + 132 + (3^{2} - 3) (4^{2} - 3)}{(3^{2} - 3) (4^{2} - 3) (5^{2} - 3)}$

Paso 2.3.11

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

$\frac{286 + 132 + (9 - 3) (4^{2} - 3)}{(3^{2} - 3) (4^{2} - 3) (5^{2} - 3)}$

Paso 2.3.12

Resta $3$ de $9$ .

$\frac{286 + 132 + 6 (4^{2} - 3)}{(3^{2} - 3) (4^{2} - 3) (5^{2} - 3)}$

Paso 2.3.13

Eleva $4$ a la potencia de $2$ .

$\frac{286 + 132 + 6 (16 - 3)}{(3^{2} - 3) (4^{2} - 3) (5^{2} - 3)}$

Paso 2.3.14

Resta $3$ de $16$ .

$\frac{286 + 132 + 6 \cdot 13}{(3^{2} - 3) (4^{2} - 3) (5^{2} - 3)}$

Paso 2.3.15

Multiplica $6$ por $13$ .

$\frac{286 + 132 + 78}{(3^{2} - 3) (4^{2} - 3) (5^{2} - 3)}$

Paso 2.4

Suma $286$ y $132$ .

$\frac{418 + 78}{(3^{2} - 3) (4^{2} - 3) (5^{2} - 3)}$

Paso 2.5

Suma $418$ y $78$ .

$\frac{496}{(3^{2} - 3) (4^{2} - 3) (5^{2} - 3)}$

Paso 2.6

Simplifica el denominador.

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Paso 2.6.1

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

$\frac{496}{(9 - 3) (4^{2} - 3) (5^{2} - 3)}$

Paso 2.6.2

Resta $3$ de $9$ .

$\frac{496}{6 (4^{2} - 3) (5^{2} - 3)}$

Paso 2.6.3

Eleva $4$ a la potencia de $2$ .

$\frac{496}{6 (16 - 3) (5^{2} - 3)}$

Paso 2.6.4

Resta $3$ de $16$ .

$\frac{496}{6 \cdot 13 (5^{2} - 3)}$

Paso 2.6.5

Multiplica $6$ por $13$ .

$\frac{496}{78 (5^{2} - 3)}$

Paso 2.6.6

Eleva $5$ a la potencia de $2$ .

$\frac{496}{78 (25 - 3)}$

Paso 2.6.7

Resta $3$ de $25$ .

$\frac{496}{78 \cdot 22}$

Paso 2.7

Multiplica $78$ por $22$ .

$\frac{496}{1716}$

Paso 2.8

Cancela el factor común de $496$ y $1716$ .

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Paso 2.8.1

Factoriza $4$ de $496$ .

$\frac{4 (124)}{1716}$

Paso 2.8.2

Cancela los factores comunes.

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Paso 2.8.2.1

Factoriza $4$ de $1716$ .

$\frac{4 \cdot 124}{4 \cdot 429}$

Paso 2.8.2.2

Cancela el factor común.

$\frac{4 \cdot 124}{4 \cdot 429}$

Paso 2.8.2.3

Reescribe la expresión.

$\frac{124}{429}$

Paso 3

El resultado puede mostrarse de distintas formas.

Forma exacta:

$\frac{124}{429}$

Forma decimal:

$0.\overline{289044}$