📚 Sequences & Series – Study Notes
(Prepared for JEECUP Mathematics)
1. Sequences
| Concept | Simple Explanation | Key Formula(s) |
|---|---|---|
| Sequence | An ordered list of numbers (a_1, a_2, a_3,\dots). Each number is called a term. | – |
| nth term | The value of the sequence at position (n). | (a_n) |
| Convergence | A sequence converges if the terms approach a fixed real number (L) as (n\to\infty). | (\displaystyle \lim_{n\to\infty} a_n = L) |
| Divergence | A sequence diverges if it does not approach any finite limit. | – |
| Monotonic | Always increasing or always decreasing. | – |
| Bounded | All terms lie between two fixed real numbers. | – |
Common Types
| Type | Definition | Formula for nth term |
|---|---|---|
| Arithmetic Sequence (AP) | Each term increases (or decreases) by a constant difference (d). | (a_n = a_1 + (n-1)d) |
| Geometric Sequence (GP) | Each term is obtained by multiplying the previous term by a constant ratio (r). | (a_n = a_1 r^{,n-1}) |
| Harmonic Sequence | Reciprocal of natural numbers. | (a_n = \dfrac{1}{n}) |
| Exponential Sequence | Terms grow like powers of a fixed base. | (a_n = c,b^n) |
| Factorial Sequence | Terms involve factorials. | (a_n = n!) |
2. Series
A series is the sum of the terms of a sequence.
[ S_n = \sum_{k=1}^{n} a_k \quad\text{(partial sum)} ]
If (\displaystyle \lim_{n\to\infty} S_n) exists, the series converges; otherwise it diverges.
Types of Series
| Series | Definition | Convergence Criterion |
|---|---|---|
| Geometric Series | (\displaystyle \sum_{k=0}^{\infty} ar^{,k}) | ( |
| p‑Series | (\displaystyle \sum_{k=1}^{\infty} \frac{1}{k^p}) | (p>1) → converges; (p\le1) → diverges |
| Harmonic Series | (\displaystyle \sum_{k=1}^{\infty} \frac{1}{k}) | Diverges (special case of p‑series with (p=1)) |
| Telescoping Series | Successive terms cancel leaving few residual terms. | Evaluate by writing partial sums, then simplify. |
| Alternating Series | (\displaystyle \sum_{k=1}^{\infty} (-1)^{k+1} b_k) where (b_k>0). | If (b_k) decreases to 0, the series converges (Leibniz test). |
Convergence Tests
| Test | When to Use | Condition |
|---|---|---|
| nth‑term test | Any series | If (\displaystyle \lim_{n\to\infty} a_n \neq 0), series diverges. |
| Comparison test | Series with positive terms | Compare to a known convergent/divergent series. |
| Limit comparison test | Positive terms | If (\displaystyle \lim_{n\to\infty}\frac{a_n}{b_n}=c>0), then (a_n) and (b_n) share convergence. |
| Ratio test | Any series | If (\displaystyle L=\lim_{n\to\infty}\left |