Pattern Recognition:

• $\frac{1}{k(k+1}$) -> 1/k - $\frac{1}{k+1}$
• $\frac{1}{k(k+2}$) -> $\frac{1}{2}$(1/k - $\frac{1}{k+2}$)
• $\frac{1}{(2k-1}$(2k+1)) -> $\frac{1}{2}$($\frac{1}{2k-1}$ - $\frac{1}{2k+1}$)
• k*k! -> (k+1)! - k!
• $\frac{1}{sqrt(k}$+sqrt(k+1)) -> sqrt(k+1) - sqrt(k)
• $\frac{1}{k(k+1}$(k+2)) -> $\frac{1}{2}$($\frac{1}{k(k+1}$) - $\frac{1}{(k+1}$(k+2)))

After decomposition: Write out first few and last few terms to identify what survives. Typically: initial terms and final terms remain.

Sum results:

• sum $\frac{1}{k(k+1}$) from 1 to n = $\frac{n}{n+1}$
• sum $\frac{1}{(2k-1}$(2k+1)) from 1 to n = $\frac{n}{2n+1}$
• sum k*k! from 1 to n = (n+1)! - 1
• sum (sqrt(k+1)-sqrt(k)) from 1 to n = sqrt(n+1) - 1

Tip: If the general term is a fraction with polynomial denominator, always try partial fractions first. If it involves square roots, try rationalization.