Series – Advanced Reasoning Ability Notes
(Suitable for MIT‑level study, yet fully self‑contained for the CUET PG exam)
1. What is a Series?
| Symbol | Meaning | Example |
|---|---|---|
| (a_n) | (n)-th term of a sequence | (a_n = 3,5,7,9,\dots) |
| (S_n) | Sum of first (n) terms | (S_5 = 3+5+7+9+11 = 35) |
| (\sum_{k=1}^{n} a_k) | Formal summation notation | (\sum_{k=1}^{4} k = 10) |
First Principles – Every series is a finite or infinite sum of the form
[ S = \sum_{k=1}^{N} a_k \quad \text{or} \quad S = \sum_{k=1}^{\infty} a_k ] where (N) may be finite or infinite and (a_k) is a real (or complex) number.
2. Classical Series Types
| Type | Definition | General Term (a_k) | Sum (S_n) |
|---|---|---|---|
| Arithmetic Series (A.P.) | Constant difference (d) | (a_k = a_1 + (k-1)d) | (S_n = \frac{n}{2}, (a_1 + a_n)) |
| Geometric Series (G.P.) | Constant ratio (r) | (a_k = a_1 r^{,k-1}) | (S_n = a_1,\frac{1-r^n}{1-r}) (if (r\neq1)) |
| Fibonacci‑type | (a_k = a_{k-1}+a_{k-2}) | (a_k = \frac{1}{\sqrt5}!\left(\phi^k-\psi^k\right)) | (S_n = a_{n+2}-1) |
| Harmonic Series | (a_k = \frac{1}{k}) | (a_k = \frac{1}{k}) | Diverges ((S_n \sim \ln n)) |
| Alternating Series | (a_k = (-1)^{k-1}b_k) | (b_k>0) decreasing | Converges if (\lim b_k=0) |
Key Insight – The general term (a_k) is the engine that drives the entire series. Once we know (a_k), all sums and properties follow.
3. Derivation of the Arithmetic Sum Formula (First Principles)
Let (a_1, a_2, \dots, a_n) be an A.P. with common difference (d).
[ S_n = a_1 + a_2 + \dots + a_n \tag{1} ]
Write the same sum in reverse order:
[ S_n = a_n + a_{n-1} + \dots + a_1 \tag{2} ]
Add (1) and (2):
[ 2S_n = (a_1+a_n) + (a_2+a_{n-1}) + \dots + (a_n+a_1) ]
Each bracket equals (a_1 + a_n). There are (n) such brackets:
[ 2S_n = n (a_1 + a_n) ;;\Rightarrow;; S_n = \frac{n}{2},(a_1+a_n) ]
4. Derivation of the Geometric Sum Formula
Let (a_1) be the first term, (r\neq1) the ratio.
[ S_n = a_1 + a_1r + a_1r^2 + \dots + a_1 r^{,n-1} ]
Multiply both sides by (r):
[ rS_n = a_1r + a_1r^2 + \dots + a_1 r^{,n} ]
Subtract:
[ S_n - rS_n = a_1 - a_1r^{,n} ;;\Rightarrow;; S_n(1-r) = a_1(1-r^n) ]
Hence
[ S_n = a_1,\frac{1-r^n}{1-r} ]
5. Edge Cases & Nuances
| Edge Case | What Happens | Notes |
|---|---|---|
| Zero ratio ((r=0)) | (S_n = a_1) | Only the first term survives. |
| Negative ratio | Alternating signs; convergence depends on ( | r |
| Common difference zero ((d=0)) | All terms equal (a_1). | (S_n = n a_1). |
| **Infinite series with ( | r | <1)** |
| **Infinite series with ( | r | \ge1)** |
| Harmonic series | Diverges (slowly). | (S_n \approx \ln n + \gamma). |
| Conditional convergence | Alternating series with decreasing ( |