telescope effect

Topics

• basic arithmetic

The telescope effect, also known as a telescoping sum, is a series where many terms cancel out, leaving only a few terms. This significantly simplifies the expression being evaluated. It’s analogous to how a telescope collapses into a much smaller, manageable size.

Explanation

The core idea behind the telescope effect is the cancellation of intermediate terms in a sum or product. This usually takes the form of:

$$(a_1-a_2)+(a_2-a_3)+(a_3-a_4)+...+(a_{n-1}-a_n)$$

Notice how the -a_2 cancels with the +a_2, -a_3 with +a_3, and so on. The only terms that survive are a_1 and -a_n, leading to a simplified result of a_1 - a_n.

Applications

• difference arrays:
  • The reconstruction of the original array from a difference array is a direct application of the telescoping effect. The intermediate terms cancel out when taking prefix sums
• hockey-stick identity:
  • The hockey-stick identity in combinatorics leverages the telescoping effect to simplify the summation of binomial coefficients
• Partial Fraction Decomposition:
  • Consider ${\sum }_{i=1}^n\frac{1}{i(i+1)}.$ Using partial fractions, the summand equals $\frac{1}{i}-\frac{1}{i+1}$
  • Thus, the sum is equal to $(1-\frac{1}{2})+(\frac{1}{2}-\frac{1}{3})+...+(\frac{1}{n}-\frac{1}{n+1})=1-\frac{1}{n+1}$

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