Sum of infinite GP — when it converges, proof

3.3.3 · Maths › Sequences & Series

Infinite GP kya hota hai?

Key word hai agar exist karta hai. Ek infinite sum ko samajhne ke liye, hum partial sums $S_{N}$ (pehle $N$ terms ka sum) dekhte hain aur poochte hain: kya yeh ek fixed number ki taraf jaate hain jab $N \to \infty$ ?

Partial-sum formula derive karna (scratch se)

KYA chahiye: $S_{N} = a + a r + \dots + a r^{N - 1}$ ke liye ek closed formula.

KAISE — shift-and-subtract trick.

$S_{N} = a + a r + a r^{2} + \dots + a r^{N - 1}$

Har term ko $r$ se multiply karo:

$r S_{N} = a r + a r^{2} + \dots + a r^{N - 1} + a r^{N}$

Yeh step kyun? $r$ se multiply karne par poori list ek jagah right shift ho jaati hai, isliye almost sab terms original ke saath align ho jaate hain — yeh cancel ho jaayenge.

Subtract karo:

$S_{N} - r S_{N} = a - a r^{N}$

Yeh step kyun? Har beech wala term dono lines mein aata hai aur cancel ho jaata hai; sirf pehla ( $a$ ) aur bahar nikla hua aakhri ( $a r^{N}$ ) bachta hai.

Factor karo aur solve karo ( $r \neq = 1$ ke liye valid):

$S_{N} (1 - r) = a (1 - r^{N}) ⟹ S_{N} = \frac{a ( 1 - r ^{N} )}{1 - r}$

Convergence condition — note ka dil

Ab $N \to \infty$ karne do. Sirf $r^{N}$ hi $N$ par depend karta hai.

Toh limit exist karta hai iff $== ∣ r ∣ < 1 ==$ , aur tab $r^{N} \to 0$ :

$S_{\infty} = lim_{N \to \infty} \frac{a ( 1 - r ^{N} )}{1 - r} = \frac{a ( 1 - 0 )}{1 - r}$

Figure — Sum of infinite GP — when it converges, proof

Worked examples

Common mistakes

Active recall

Recall Test yourself (answers chhupao)

Q: Kaun si ek condition guarantee karti hai ki infinite GP ka finite sum hoga? A: $∣ r ∣ < 1$ .
Q: Formula $\frac{a}{1 - r}$ tak kyun collapse hota hai? A: Kyunki $r^{N} \to 0$ hota hai $S_{N} = \frac{a ( 1 - r ^{N} )}{1 - r}$ mein.
Q: $\sum_{n = 0}^{\infty} (1/3)^{n}$ kya hai? A: $\frac{1}{1 - 1/3} = \frac{3}{2}$ .
Q: Kya $1 - 1 + 1 - 1 + \dots$ converge karta hai? A: Nahi; $r = - 1$ , partial sums $1, 0, 1, 0$ oscillate karte hain.

Recall Feynman: ek 12-saal ke bacche ko samjhao

Socho ek chocolate bar hai. Tum aadha khaate ho, phir jo bacha uska aadha, phir uska aadha... Tum hamesha ke liye chhote chhote tukde khaate rehte ho. Total mein kitni chocolate khaoge? Poora bar — exactly 1. Bhale hi infinitely many bites hain, lekin woh itne tiny ho jaate hain ki sab kuch ek finite amount mein add ho jaata hai. Yeh tabhi kaam karta hai jab har bite pichle se chhota ho (ratio $< 1$ ). Agar har bite bada hota, toh kabhi ruko hi nahi aur koi total banta hi nahi.

Connections

Geometric Progression — nth term
Sum of finite GP
Limits of Sequences (kyun $r^{N} \to 0$ )
Convergence of Series
Recurring Decimals as Fractions
Present Value & Perpetuities (finance application)

The infinite GP

\sum a r^{n}

converges iff

∣ r ∣ < 1

Sum of convergent infinite GP

S_{\infty} = \frac{a}{1 - r}

for

∣ r ∣ < 1

Closed form for partial sum

S_{N}

of a GP

S_{N} = \frac{a ( 1 - r ^{N} )}{1 - r}

(for

r \neq = 1

)

Trick used to derive

S_{N}

r

se multiply karo, shift karo, aur subtract karo taaki beech ke terms cancel ho jayein

Why

S_{\infty} = \frac{a}{1 - r}

needs

∣ r ∣ < 1

Tabhi

r^{N} \to 0

hota hai, jo

r^{N}

wale term ko khatam karta hai

Value of

0. \overline{3}

as GP sum

\frac{0.3}{1 - 0.1} = \frac{1}{3}

Sum

2 + 1 + \frac{1}{2} + \dots

4

Does

1 + 2 + 4 + 8 + \dots

have a finite sum?

Nahi,

r = 2 \geq 1

, yeh diverge karta hai

What happens to partial sums when

r = - 1

Woh oscillate karte hain aur kabhi settle nahi hote