Pattern Recognition:

• 1/(k(k+1)) -> 1/k - 1/(k+1)
• 1/(k(k+2)) -> (1/2)(1/k - 1/(k+2))
• 1/((2k-1)(2k+1)) -> (1/2)(1/(2k-1) - 1/(2k+1))
• k*k! -> (k+1)! - k!
• 1/(sqrt(k)+sqrt(k+1)) -> sqrt(k+1) - sqrt(k)
• 1/(k(k+1)(k+2)) -> (1/2)(1/(k(k+1)) - 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 1/(k(k+1)) from 1 to n = n/(n+1)
• sum 1/((2k-1)(2k+1)) from 1 to n = 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.