4.3.10 · D1Alternating series test — Leibniz test, proof
Agar aap baar baar aisa step lete ho jo har baar direction badle aur zero ki taraf chhota hota jaye , toh aap forever bhatakne wale nahi ho — aapko ek jagah settle hona hi padega. Poora alternating series test bas yahi "flag ki taraf hop karna" wali picture hai, exact banai gayi, aur neeche sab kuch woh vocabulary hai jo isse precisely kehne ke liye chahiye.
Yeh page assume karta hai ki aapko kuch nahi pata . Har symbol jo parent proof use karta hai, woh yahan banaya gaya hai, ek ke upar doosra, aur har ek ke saath ek picture.
Signs aur limits se pehle, humein raw object chahiye: ek sum jo kabhi khatam nahi hota .
Definition Sequence vs. seriesEk sequence numbers ki ek ordered list hai, har counting number ke liye ek: a 1 , a 2 , a 3 , … series woh hai jo aapko milta hai jab aap sequence ke terms ko add karo: a 1 + a 2 + a 3 + ⋯
Picture: sequence number line par dots hain (har step ke liye ek); series woh running total hai jab aap un dots se guzarte ho.
Intuition Kyun hum "sab add" nahi kar sakteAap literally finite time mein infinitely many numbers add nahi kar sakte. Toh hum cheat karte hain: hum pehle N add karte hain, us running total ko dekhte hain, aur poochhte hain "yeh kahan ja raha hai?" Woh heading hi poora game hai. Dekho Convergence of Series — overview .
n n counter hai: yeh 1 , 2 , 3 , … kaunsa term mean kar rahe hain. a n n
Picture: n step number hai. a 5
Topic ko yeh kyun chahiye: bina counter ke hum yeh nahi keh sakte "is term ke baad wala term" (a n + 1 n → ∞ n
Definition Sigma notationn = 1 ∑ N a n a 1 + a 2 + ⋯ + a N ∑ add karo ; neeche n = 1 1 se count shuru karo ; upar N N ∞
Picture: ∑ machine hai jisme start dial (neeche) aur stop dial (upar) hai jo har term ko ek running total mein daalti hai.
∑ n = 1 ∑ ∞ process hai, automatically ek value nahi. Kya yeh banta hai ek number — yahi "converges" ka matlab hai.
Yeh proof mein sabse important object hai, isliye hum isko apni alag picture dete hain.
s N s N = ∑ n = 1 N a n = a 1 + a 2 + ⋯ + a N . N baad ka total hai — ek normal, finite sum. s 1 = a 1 s 2 = a 1 + a 2
Picture: number line par har step ke baad apni position mark karo. s 1 , s 2 , s 3 , … running totals se bani ek nayi sequence .
Topic ko yeh kyun chahiye: parent proof kabhi bhi directly infinite sum ko study nahi karta. Woh partial sums ke sequence s N woh positions settle hoti hain. Series ka convergence defined hai as convergence of s N
− 1 ( − 1 ) + 1 − 1 ( − 1 ) 0 = + 1 , ( − 1 ) 1 = − 1 , ( − 1 ) 2 = + 1 , ( − 1 ) 3 = − 1 , … Even power ⟹ + 1 − 1 (− 1
Ab ( − 1 ) n − 1
n n − 1 ( − 1 ) n − 1
1
0
+ 1
2
1
− 1
3
2
+ 1
4
3
− 1
Toh ( − 1 ) n − 1 + , − , + , − , … positive se shuru karke. Isliye parent likhta hai
∑ ( − 1 ) n − 1 b n = b 1 − b 2 + b 3 − b 4 + ⋯
Intuition Sign ko size se alag kyun kareinHar term ko sign ( − 1 ) n − 1 positive size b n saari conditions b n
Definition Absolute value∣ x ∣ ∣ x ∣ x 0 ≥ 0 ∣ − 3 ∣ = 3 ∣3∣ = 3
Picture: ∣ x ∣ x
b n b n = ∣ a n ∣ > 0 magnitude hai — hop ki length , direction bhoolke. Ek alternating series mein a n = ( − 1 ) n − 1 b n
Topic ko yeh kyun chahiye: test ki dono conditions ("decreasing" aur "goes to zero") dono hop lengths ke baare mein hain, toh humein un lengths ke liye akele ek naam chahiye: woh naam hai b n
Definition Monotone decreasingEk sequence decreasing (monotone non-increasing) hai agar har term pehle wale se badi nahi ho : b n + 1 ≤ b n n
Picture: dots downhill (ya flat) chalte hain, kabhi uphill nahi . Walk mein har hop pichle se chhhota ya barabar hai.
Intuition "Decreasing" kyun dono conditions mein se ek haiProof mein, terms ko ( b 1 − b 2 ) + ( b 3 − b 4 ) + ⋯ har bracket ≥ 0 . Yeh tab sach hai jab exactly b 1 ≥ b 2 b 3 ≥ b 4 Monotone Convergence Theorem .
Common mistake "Eventually decreasing" kaafi haiCondition sirf bade n ke liye hold karni chahiye (kisi finite point ke baad). Kuch early wiggles matter nahi karti — tail hi convergence determine karta hai.
Yeh doosri condition hai, aur ∞
Definition Limit equals zero (seedhe alfaazon mein)n → ∞ lim b n = 0 jaise jaise n b n 0 Koi bhi tiny target distance chuno; eventually har b n
Picture: hop lengths zero ki taraf shrink hoti hain — 0
Intuition "Zero ko jaata hai" kyun doosri condition haiStep 4 mein proof likhta hai s 2 m + 1 = s 2 m + b 2 m + 1 b 2 m + 1 → 0 ek hi jagah milne ko majboor hain. Agar hops zero tak nahi shrink hote, toh even aur odd totals permanent gap par reh sakte hain — koi single limit nahi, koi convergence nahi.
Common mistake "Zero ko jaata hai" akele kaafi nahi hai (generally)Harmonic Series ∑ 1/ n → 0 diverge karta hai. Vanishing terms zaroori hain lekin sufficient nahi — aapko alternation aur decrease bhi chahiye. Yahi hai jo nth-Term Test for Divergence aur parent mein Mistake A warn karta hai.
Definition Series convergesSeries ∑ a n s N → ∞ lim s N = s s sum kehlata hai. Agar aisa koi number exist nahi karta, toh series diverge karti hai.
Picture: running-total dots s 1 , s 2 , s 3 , … ek point s — woh flag jis ki taraf aap hop kar rahe the.
Topic ko yeh kyun chahiye: Leibniz test ka poora conclusion ek word hai — converges . Ab aapko pata hai ki iska matlab hai "partial-sum positions ek jagah settle hoti hain," aur aap stronger cousin se miloge Absolute vs Conditional Convergence mein.
Converges means sN settles
Har sawaal padho, zyaban se jawab do, phir reveal karo.
a n n a n n
n = 1 ∑ N a n a 1 + a 2 + ⋯ + a N n = 1 n = N
Partial sum s N s N = a 1 + ⋯ + a N N s N
n = 1 , 2 , 3 , 4 ( − 1 ) n − 1 + 1 , − 1 , + 1 , − 1
∣ − 7 ∣ ∣ x ∣ 7 x 0
"Sizes decreasing hain" ko inequality mein likho. b n + 1 ≤ b n n
Plain words mein batao n → ∞ lim b n = 0 Jaise jaise n b n 0
Kya "terms 0 Nahi — jaise harmonic series ∑ 1/ n 1/ n → 0
Kiska matlab hai ki ek series s Iske partial sums satisfy karte hain lim N → ∞ s N = s s
