Infinite GP convergence: sum = $\frac{a}{1-r}$ converges iff |r| < 1.

Common infinite series in JEE:

• 1 + 1/2 + 1/4 + 1/8 + ... = 2 (GP with r=1/2)
• 1 - 1/2 + 1/4 - 1/8 + ... = 2/3 (GP with r=-1/2)
• 1 + 1/3 + 1/9 + 1/27 + ... = 3/2 (GP with r=1/3)
• 0.999... = 9/10 * $\frac{1}{1-1/10}$ = 1

Infinite AGP: S = $\frac{a}{1-r}$ + $\frac{dr}{1-r}$^2 when |r| < 1.

Recurring decimals as fractions: 0.abcabc... = abc/999 0.ababab... = ab/99 0.aaa... = a/9

Warning: Not all series are GP. If a problem gives a series, verify the ratio is constant before using GP formulas.