4.3.3 · D5 · HinglishCalculus III — Sequences & Series
Question bank — Series — partial sums, convergence definition
4.3.3 · D5· Maths › Calculus III — Sequences & Series › Series — partial sums, convergence definition
Shuru karne se pehle, teen words jo tumhare dimaag mein bilkul clear hone chahiye (har ek parent note mein fully banaya gaya hai):
- term — us list mein position par individual number jise tum add kar rahe ho.
- partial sum — terms ke baad running total.
- series converges — matlab running totals ki sequence ka ek finite limit hai.
Teeno ko alag rakho: terms, running totals, aur running totals ka limit. Neeche ka almost har trap inhi teen mein se ek ko doosre ke saath quietly swap karna hai.
True or false — justify
A term list guarantees convergence
False — Harmonic series mein hai phir bhi uske running totals bina bound ke badhte hain. Chhote terms bhi infinity tak accumulate ho sakte hain agar woh bahut slowly shrink karein.
If a series converges then its terms must
True — kyunki aur dono , isliye terms ki taraf jaate hain. Yahi n-th Term Test ki sahi direction hai.
"Diverges" always means the running total marches off to
False — ke partial sums hain jo bounded rehte hain lekin kabhi ek number par settle nahi karte, isliye yeh infinity pe gaye bina diverge karta hai.
holds for every
False — yeh formula sirf tab valid hai jab ho, kyunki yeh se born hua hai, jo hone par fail ho jaata hai.
The sequence aur series basically same object hain
False — terms ki list hai; ka jiyo ya maro running totals ki alag sequence ke through hota hai. Dono opposite behave kar sakte hain (jaise but diverge kare).
Removing or changing the first ten terms ek divergent series ko convergent bana sakta hai
False — ek finite change sirf har baad ke partial sum ko ek fixed constant se shift karta hai, jo ek limit create ya destroy nahi kar sakta. Convergence ek "tail" property hai.
Agar ek bounded sequence hai, toh series converge karti hai
False — bounded, convergent ke barabar nahi hai; ke bounded partial sums () kabhi ek single value ki taraf approach nahi karte.
Ek telescoping series hamesha converge karti hai
False — telescoping sirf ke liye ek clean closed form deta hai; yeh converge karega ya nahi yeh abhi bhi is baat par depend karta hai ki us closed form ka finite limit hai ya nahi. Dekho Telescoping series.
Agar do series dono diverge karti hain, toh unka sum zaroor diverge karega
False — lo aur ; har ek diverge karta hai lekin combined series ke saare terms zero hain aur yeh par converge karta hai.
Spot the error
", therefore converges." Galti kahan hai?
N-th Term Test sirf ek direction mein chalta hai: convergence force karti hai ki ho, lekin convergence force nahi karta. Harmonic series iska standing counterexample hai.
"." Kya galat hua?
Geometric formula ke saath apply kiya gaya, lekin hai, isliye series actually diverge karti hai — koi finite sum nahi hai, aur ek meaningless plug-in hai.
"The series equals the sum , so I'll just add them all up." Yeh definition kyun nahi hai?
Addition ek binary operation hai — tum ek waqt mein sirf do numbers add kar sakte ho — isliye "infinitely many add karo" undefined hai. Real definition ke through jaati hai.
" ka matlab hai ki kisi point se aage har partial sum ke equal ho jaata hai." Sahi hai?
Nahi — -definition kehta hai ki har partial sum eventually ke half-width ke band ke andar land karta hai, exactly par nahi. Woh approach karte hain, unhe kabhi pahunchna zaroori nahi.
"Kyunki harmonic series ka har block thoda se zyada hai aur infinite blocks hain, isliye sum hai." Kya logic valid hai?
Haan — infinite baar se zyada add karne se kisi bhi bound se zyada ho jaata hai, jo exactly ek divergent (unbounded) running total hai. Yeh wala correct reasoning hai, trap yeh hai ki isko doubt karo.
" diverge karta hai kyunki uske terms nahi jaate." Sahi conclusion, lekin reason check karo.
Conclusion sahi hai aur reason bhi sahi hai: oscillate karta hai aur kabhi approach nahi karta, isliye n-th Term Test isko khatam karta hai. Trap yeh realize karna hai ki ek bounded series phir bhi term test fail kar sakti hai.
Why questions
Hum convergence define kyon partial sums ke through karte hain instead of raw infinite sum ke?
Kyunki "forever add karo" ek legal operation nahi hai, jabki running totals ki ek sequence ya toh limit rakhti hai ya nahi — ek impossible task ko us cheez mein convert karna jise hum Sequences — limits and convergence se pehle hi solve kar chuke hain.
ko se multiply karke subtract karna geometric partial sum ko kyon collapse karta hai?
Factor har term ko ek slot right shift karta hai, isliye saare middle terms line up ho jaate hain aur cancel ho jaate hain, sirf pehla term aur aakhiri leftover bachta hai — ek telescoping cancellation.
Geometric convergence ke liye exactly border kyun hai?
Kyunki ka sirf -dependent piece hai, aur precisely tab hota hai jab ; par yeh shrink nahi karta, aur par yeh blow up karta hai.
N-th Term Test sirf divergence ka test kyun hai aur kabhi convergence ka nahi?
Iska ek hi guarantee contrapositively chalta hai — divergence prove karta hai — lekin outcome ko open chhod deta hai, isliye yeh kabhi convergence certify nahi kar sakta.
Ek single bahut bada lekin finite partial sum humein kyun nahi bata sakta ki series diverge karti hai?
Convergence ke tail behaviour ke baare mein hai jab ; koi bhi finite sirf ek snapshot hai, aur running total abhi bhi uske kaafi aage settle ho sakta hai.
Parent note kyun insist karta hai "hamesha ke through jao"?
Kyunki do sabse common errors dono terms ke baare mein directly reason karne se aate hain, jabki series ka fate poori tarah running totals ke limit se decide hota hai.
Edge cases
Jab ho toh ka kya hota hai?
Har term ke barabar hai, isliye , jo tak march karta hai ( ke liye) — series diverge karti hai, isliye formula ko exclude karta hai.
Jab ho toh ka kya hota hai?
Partial sums bounce karte hain aur kabhi settle nahi karte, isliye yeh diverge karta hai — bounded oscillation, blow-up nahi.
Jab ho (har term zero) toh sum kya hai?
Saare partial sums hain, isliye series trivially par converge karti hai ki parwah kiye bina — ek degenerate lekin valid convergent case.
Kya harmonic series kabhi "stop" badhna karta hai kyunki terms microscopic ho jaate hain?
Nahi — halanki , partial sums eventually har finite bound se zyada ho jaate hain; terms shrink karte hain lekin itna fast nahi ki total cap ho sake.
Kya ek series converge kar sakti hai agar infinitely many terms exactly hain?
Haan — zero terms running total mein kuch add nahi karte, isliye convergence sirf nonzero terms par depend karta hai; jaise of surviving p-series ki tarah behave karta hai.
Agar , toh kya yeh "ek limit" hai?
Convergence ke sense mein, nahi — ek series sirf ek finite number par converge karti hai, isliye ek infinite limit divergence count hoti hai chahe ka ek definite (infinite) trend ho.
Connections
- Parent: partial sums & convergence
- n-th Term Test for Divergence — yahan sabse bade trap ka source
- Harmonic series — " converges" ka eternal counterexample
- Geometric series — jahan passport traps rehte hain
- Telescoping series — clean limit guarantee nahi karta
- Sequences — limits and convergence — har trap ek sequence limit tak reduce hota hai