Telescoping, Sigma & Induction

Why telescoping matters

Many HSC series are not pure APs or GPs. After a clever rewrite, consecutive terms cancel and only the “ends” survive. That pattern is called telescoping. Typical tools:

1. Partial fractions: $\dfrac{1}{k(k+1)}=\dfrac{1}{k}-\dfrac{1}{k+1}$.
2. Difference of surds: rationalise to expose cancellation.
3. Induction: prove the guessed closed form for every $n$.

Worked example 1 — Classic partial fractions

Evaluate $\displaystyle\sum_{k=1}^{n}\frac{1}{k(k+1)}$, then find the infinite sum.

Decompose:

$$\frac{1}{k(k+1)}=\frac{1}{k}-\frac{1}{k+1}.$$

The partial sum collapses:

$$\sum_{k=1}^{n}\left(\frac{1}{k}-\frac{1}{k+1}\right)=1-\frac{1}{n+1}=\frac{n}{n+1}.$$

As $n\to\infty$, $S_\infty=1$.

Induction sketch: assume $S_n=\frac{n}{n+1}$; then $S_{n+1}=S_n+\frac{1}{(n+1)(n+2)}=\frac{n}{n+1}+\frac{1}{(n+1)(n+2)}=\frac{n+1}{n+2}$.

Worked example 2 — Surd telescoping

Sums involving $\sqrt{k+1}-\sqrt{k}$ (or conjugates) telescope after rationalisation. Multiply by the conjugate to turn a difference of square roots into a difference of integers or simpler radicals, then cancel.

Strategy: write the general term as $c\bigl(f(k)-f(k+1)\bigr)$ or $c\bigl(f(k)-f(k-1)\bigr)$, identify $f$, and evaluate the leftover boundary terms.

Worked example 3 — Alternating or shifted patterns

Some series need a non-obvious decomposition (e.g. three-term partial fractions, or pairing $a_k-a_{k+2}$). Steps:

1. Guess the telescoping shape from small $n$.
2. Prove the identity algebraically.
3. Sum from $k=1$ to $n$ and simplify.
4. Take limits if asked.

Index shifts in sigma notation often appear here — rewrite $\sum_{k=2}^{n+1} a_{k-1}$ back to a standard range before cancelling.

Link to binomial and partial sums

Partial sums of binomial coefficients and related identities sometimes telescope after using Pascal’s relation $\binom{n}{k}=\binom{n-1}{k}+\binom{n-1}{k-1}$. Treat those as “structured telescoping” once the AP/GP basics are fluent.

Common mistakes

• Cancelling terms that are not adjacent after the rewrite.
• Forgetting the last uncancelled positive/negative boundary term.
• Claiming $S_\infty$ without checking the remainder $\to 0$.
• Induction: proving the step but skipping a clear base case $n=1$.

Practice (hints only)

1. Find $\sum_{k=1}^{n}\frac{1}{k(k+2)}$. (Partial fractions with three terms or careful pairing.)
2. Prove by induction a closed form you obtain by telescoping, then let $n\to\infty$.

Where next

Recursive sequences and limits build on the same careful algebra: see recurrence, limits, and iteration.