Express rational functions as a sum of fractions with simpler denominators. You can apply this to telescoping series to mass cancel terms in a seemingly complicated sum.

By the coverup rule, the coefficient of the term \( \frac{1}{ x + 6 } \) in the partial fraction decomposition of

\[ \frac{ 1} { ( x + 6 ) ( x + 19 ) },\]

has the form \( \frac{1}{A} \). What is \(A\)?

By the coverup rule, the coefficient of the term \( \frac{1}{ x + 2 } \) in the partial fraction decomposition of

\[ \frac{ 1} { ( x - 2 ) ( x + 2 ) ( x + 7 ) },\]

has the form \( \frac{1}{A} \). What is \(A\)?

By the coverup rule, the coefficient of the term \( \frac{1}{ x + 4 } \) in the partial fraction decomposition of

\[ \frac{ 1} { ( x + 4 ) ( x^2 + 5 x + 20 ) },\]

has the form \( \frac{1}{A} \). What is \(A\)?

By the coverup rule, the coefficient of the term \( \frac{1}{ x + 2 } \) in the partial fraction decomposition of

\[ \frac{ 1} { ( x + 2 ) ( x^2 + 4 x + 28 ) },\]

has the form \( \frac{1}{A} \). What is \(A\)?

By the coverup rule, the coefficient of the term \( \frac{1}{ x + 3 } \) in the partial fraction decomposition of

\[ \frac{ 1} { ( x + 3 ) ( x + 9 )(x+16) },\]

has the form \( \frac{1}{A} \). What is \(A\)?

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