Sigma (Σ) notation — evaluating, telescoping sums

3.3.7 · Maths › Sequences & Series

Sigma notation KYA hai?

"+1" KYUN? $k = 2, 3, 4, 5$ count karo: kya ye $5 - 2 = 3$ hai? Nahi — ye $4$ numbers hain. Endpoints ghataane se hamesha ek kam ho jaata hai, isliye usse wapas jodte hain: $n - m + 1$ .

Teen algebra rules (YE KYUN kaam karte hain)

KYUN? Addition commutative aur associative hai — ek finite sum ko aap freely reorder aur regroup kar sakte ho. $c$ bahar nikalana sirf distributive law hai jo $n$ baar apply hota hai. Aur ek constant $c$ jo $n$ baar add ho, wo $n c$ hai (kyunki $n$ terms hain, har ek $c$ ke barabar hai).

Standard formulas scratch se derive karna

$\sum k^{2}$ KAISE derive karte hain (cubes ko telescope karna — ek preview!): Identity $(k + 1)^{3} - k^{3} = 3 k^{2} + 3 k + 1$ use karo. Dono sides ko $k = 1$ se $n$ tak sum karo. Left side telescope ho jaati hai (agli section dekho) $(n + 1)^{3} - 1$ mein: $(n + 1)^{3} - 1 = 3 \sum k^{2} + 3 \sum k + \sum 1 = 3 \sum k^{2} + 3 \cdot \frac{n ( n + 1 )}{2} + n .$ $\sum k^{2}$ ke liye solve karo aur simplify karo → $\frac{n ( n + 1 ) ( 2 n + 1 )}{6}$ . Ye trick KYUN? Kyunki cube-difference $k^{2}$ ke baare mein jaanta hai, aur ek difference ko sum karna cleanly collapse ho jaata hai.

Telescoping — show ka star

Figure — Sigma (Σ) notation — evaluating, telescoping sums

Telescoping sum KAISE pehchante hain: general term aksar ek aisi fraction hoti hai jiska denominator consecutive-ish pieces mein factor ho jaata hai, jaise $\frac{1}{k ( k + 1 )}$ . Isse $f (k) - f (k + 1)$ mein split karne ke liye partial fractions use karo.

Worked examples

Recall Feynman: 12-saal ke bachche ko samjhao

Ek lambi line mein khade dominoes imagine karo. Sigma bas kehta hai "line mein sab kuch count karo." Ek telescoping sum ek magical line hai jahan har domino ek plus hai aur bilkul agli same number ka minus hai — to wo ek doosre ko knock out kar dete hain! Saari cancelling ke baad, sirf pehla aur aakhri domino khade rahte hain. To sau cheezein add karne ki jagah, tum bas "pehla minus aakhri" karte ho. Yehi saara trick hai.

Flashcards

\sum_{k = m}^{n} a_{k}

ka matlab kya hai?

a_{m} + a_{m + 1} + \dots + a_{n}

— general term ko har integer

k

ke liye

m

n

tak add karo.

\sum_{k = m}^{n}

mein kitne terms hain?

n - m + 1

\sum_{k = 1}^{n} k

batao.

\frac{n ( n + 1 )}{2}

(Gauss: aage+peeche pair karo,

2 S = n (n + 1)

\sum_{k = 1}^{n} k^{2}

batao.

\frac{n ( n + 1 ) ( 2 n + 1 )}{6}

\sum_{k = 1}^{n} k^{3}

batao.

[\frac{n ( n + 1 )}{2}]^{2}

(integers ke sum ka square).

\sum (c a_{k}) = c \sum a_{k}

KYUN hota hai?

Distributive law:

c

har term se bahar aa jaata hai.

Kya

\sum a_{k} b_{k} = (\sum a_{k}) (\sum b_{k})

hai?

Nahi — products Σ par distribute nahi hote (counterexample:

k \cdot k

General telescoping result kya hai?

\sum_{k = m}^{n} [f (k) - f (k + 1)] = f (m) - f (n + 1)

\frac{1}{k ( k + 1 )}

ko telescoping form mein kaise badate ho?

Partial fractions:

\frac{1}{k} - \frac{1}{k + 1}

\sum_{k = 1}^{n} \frac{1}{k ( k + 1 )}

evaluate karo.

1 - \frac{1}{n + 1} = \frac{n}{n + 1}

Gap-2 sums jaise

\frac{1}{k ( k + 2 )}

har end par do survivors kyun chhorte hain?

Cancellation ek term se delay hoti hai, isliye do heads aur do tails bachti hain.

Connections

Arithmetic Progressions — $\sum (a + (k - 1) d)$ sirf linearity + $\sum k$ hai.
Partial Fractions — telescoping structure banana ka key tool.
Method of Differences — telescoping iska finite-sum incarnation hai.
Mathematical Induction — standard Σ formulas ko rigorously prove karne ke liye use hota hai.
Convergence of Series — telescoping exact partial sums deta hai, to limits aasaan ho jaati hain.
Binomial Theorem — expansions Σ ke saath compactly likhte hain.