### Shortcut Methods

# **Infinite Series Tricks and Shortcuts: JEE & CBSE Solutions **

**1. Infinite Geometric Series Sum:**

Formula: $S = \frac{a}{1-r}$ where ‘a’ is the first term , ‘r’ is common ratio , ‘S’ denotes the sum

##**2. Ratio Test (Convergence/Divergence):**
If $\lim\limits_{n\to\infty} \left| \frac{a_{n+1}}{a_n} \right| = L$

- Converges if $ L < 1 $ *Diverges if $ L > 1$

*Ambiguous (inconclusive) when $ L = 1 $ > >

**3. Alternating Series Approximation Error:**

$$\text{Error} \leq b_{n+1}$$

where n is an index large enough, and ({ b_{n+1} }) is the succeeding term after the final considered term .

These methods significantly simplify complex sums, enable effective convergence, divergence analysis, and provide accurate numeric estimations for JEE and CBSE examinations!