Foundations — Geometric series — convergence condition, proof

4.3.4 · D1 · Maths › Calculus III — Sequences & Series › Geometric series — convergence condition, proof

Yeh page parent note Geometric series mein use hone wale har symbol aur idea ko unpack karta hai, un cheezoon se shuru karke jo ek samajhdar 12-saal-ka bachcha pehle se jaanta hai. Yahan yeh assume nahi kiya ki aap "series", "sigma", ya "limit" se pehle mil chuke ho — hum har cheez ek picture se banate hain.

---

1. Pehla term $a$, aur ek fixed factor se multiply karna

"Series" se pehle, sabse simple object numbers ki ek list hai jahan har number pichle se ek hi number se multiply karke aata hai. Aisi list describe karne ke liye hamen sirf do cheezein chahiye: shuruat kahan hoti hai aur kisse multiply hota hai.

Figure dekho. Sabse baayaan bar pehla term $a$ hai; har agla bar apne baayein wale bar ki height ka $r$ guna hai. Jab $r$ $0$ aur $1$ ke beech hota hai, bars shrink hote hain — yahi shrinking convergence ki poori kahani hai.

---

2. Powers aur notation $r^n$

$r\cdot r\cdot r$ likhna thakaa deta hai, isliye hum ek exponent use karte hain.

Jo cheez logon ko trip karti hai: $==r^0=1==$. Kyun? $r$ ki zero copies ko saath multiply karna "empty product" hai, aur kuch bhi multiply nahi karna aapko multiplication ke starting point par chhodta hai, jo $1$ hai. Yeh matter karta hai kyunki $n=0$ se shuru hone wali series ka pehla term $a\cdot r^0=a$ hota hai.

---

3. Absolute value $|r|$ — ek number ki size

Convergence rule $|r|$ ke saath state hoti hai, $r$ ke saath nahi. Hamen yeh isliye chahiye kyunki negative $r$ bhi pieces ko shrink kara sakta hai.

Figure $r$ ko ek horizontal number line (woh axis jo zero se left-right mein jaati hai) par rakhta hai. Blue double-arrow $0$ se $-3$ tak ki distance measure karta hai, aur orange double-arrow $0$ se $+3$ tak. Dono arrows ki same length hai, $3$ — yeh exactly $|-3|=|3|=3$ ki picture hai: arrows opposite directions mein point karte hain, lekin sirf unki length hi woh hai jo $|r|$ report karta hai.

---

4. Summation symbol $\sum$

Ab woh piece jo parent note har jagah use karta hai.

Poora likha hua, wohi sum hai ${\sum }_{n=0}^{N-1}a{r}^n=\underset{n=0}{\underbrace{a{r}^0}}+\underset{n=1}{\underbrace{a{r}^1}}+\underset{n=2}{\underbrace{a{r}^2}}+\cdots +\underset{n=N-1}{\underbrace{a{r}^{N-1}}}.$