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4.3.19 · D5 · HinglishCalculus III — Sequences & Series

Question bank — Applications — approximation, evaluating limits

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4.3.19 · D5 · Maths › Calculus III — Sequences & Series › Applications — approximation, evaluating limits

Questions se pehle, ek chhoti si glossary taaki neeche har symbol samajh aaye:

True ya false — justify karo

TF1. "Ek Taylor polynomial , ke exactly equal hota hai."
False — yeh plus remainder ke equal hota hai; sirf poori infinite series (apne radius ke andar) ke equal hoti hai. ek approximation hai, aur us gap ko measure karta hai.
TF2. "Taylor polynomial mein aur terms add karne se par value hamesha zyada accurate hoti hai."
Sirf radius of convergence ke andar. ke bahar terms shrink hone ki jagah grow karte hain, isliye "zyada terms" use karna ise bura banata hai, behtar nahi.
TF3. " ki Maclaurin series mein sirf even powers hain kyunki ek even function hai."
True — ek even function ke odd derivatives par vanish ho jaate hain, isliye har odd coefficient zero hota hai, sirf bachta hai.
TF4. "Lagrange remainder tumhe exact error batata hai."
False — yeh ek exact form naam karta hai, lekin unknown hai, isliye practice mein tum sirf ek bound paate ho ko uske max se replace karke.
TF5. " ke liye, ke baad error at most hai."
True — ki Maclaurin series decreasing terms ke saath alternate karti hai, isliye alternating-series bound se error pehla omitted term hai.
TF6. "Tum series ko par use kar sakte ho."
Endpoint par True — interval hai, aur Alternating Series Test se converge karta hai ( par), chahe na kare.
TF7. "Ek bada radius of convergence matlab fixed number of terms ke liye ek zyada accurate approximation hai."
False — radius decide karta hai kahan series kaam karti hai bilkul, na ki kitni tezi se wahan converge karti hai. Kisi point par accuracy aur se govern hoti hai, se nahi.
TF8. "Kyunki ki Maclaurin series sabhi ke liye converge karti hai, ke liye do terms kaafi hain."
False — convergence eventually guaranteed hai, lekin bade ke liye tumhe bahut terms chahiye; kahi bhi ke paas nahi hai.

Error dhundho

SE1. ": plug karo mein jo deta hai, isliye limit hai."
Error — tumne cancel karne se pehle plug in kiya. Series rakho: , se divide karo, phir let karo taaki mile.
SE2. " dhundhne ke liye, maine ko sirf first order tak expand kiya: . Phir numerator , limit ."
Error — sirf order tak expand karna exactly woh term phenk deta hai jisko tumhe chahiye. Denominator order ka hai, isliye ko tak rakho; bachne wala leading term deta hai.
SE3. ", aur , isliye ratio hai."
Error — ek ratio ka matlab divide karna hai, multiply nahi: . Saath hi ka sign negative hai.
SE4. " ko par equals karta hai."
Error — series ke liye diverge karti hai, isliye woh terms kuch bhi meaningful nahi add karte. paane ke liye tumhe recentre karna hoga ya ek convergent rearrangement use karna hoga; Radius & Interval of Convergence dekho.
SE5. "Maine ki error ko se bound kiya kyunki on ."
Error — par, increase hoke tak jaata hai, isliye false hai. Ek safe bound hai; ek bound par ek upper limit hona chahiye.
SE6. " ke liye maine use kiya, isliye numerator , limit ."
Error — ka term exactly wahi hai jo bachta hai: , isliye se divide karne par milta hai. Hamesha denominator ke order tak expand karo.
SE7. "Alternating-series error bound par apply hoti hai."
Error — us series ke saare terms positive hain, isliye yeh alternate nahi karta; alternating bound invalid hai. Iske bajaaye Lagrange remainder use karo.

Why questions

WHY1. limits ke liye series substitute karna aksar L'Hôpital's Rule se better kyun hota hai?
Series saari "differentiating" ek baar, pehle se, coefficients ke roop mein kar leti hai; tum lowest power ko ek hi step mein cancel karte ho, baar baar quotient/derivative rule apply karne ki jagah.
WHY2. Ek limit mein tumhe terms at least denominator ki degree tak kyun rakhne chahiye?
Lower powers zero par cancel ho jaate hain, isliye jawab us term se decide hota hai jiski order denominator se match kare; use phenk do aur poori limit kho doge.
WHY3. Taylor coefficient ke denominator mein factorial kyun hai?
ko exactly times differentiate karne par produce hota hai; se divide karne par yeh cancel ho jaata hai taaki ki -th derivative reproduce kare aur kuch nahi — derivation Taylor & Maclaurin Series mein dekho.
WHY4. Leading (lowest-power) term par dominate kyun karta hai jab yeh "chhota" lagta hai?
ke liye, ek chhota exponent ek bada number deta hai (), isliye sabse chhoti power sabse badi hai; higher-order terms tezi se vanish ho jaate hain aur negligible ban jaate hain.
WHY5. Ek chhota ek fixed-order approximation ko behtar kyun banata hai?
Error ke jaisi scale hoti hai; us doori ko chhota karna raised power ko dramatically chhota karta hai, kyunki ek chhote number ko badi power tak uthane par woh zero ki taraf jaata hai.
WHY6. "disguise mein agla term" kyun hai?
Iska shape exactly -th Taylor term jaisa hai, lekin derivative par evaluate hone ki jagah ek unknown interior point par evaluate hoti hai — polynomial jo kuch chhhod gayi sab kuch capture karta hai; Lagrange Remainder & Error Estimation dekho.
WHY7. Tum radius ke andar power series ko term by term add, multiply, aur differentiate kyun kar sakte ho?
ke andar series absolutely aur uniformly converge karti hai taaki ye operations polynomials ki tarah behave karein — justification Power Series — Operations (add, multiply, differentiate) mein hai.
WHY8. Parent note polynomial ko par "best" fit kyun kehta hai?
Kyunki uske coefficients woh unique choice hain jo ki value aur par har derivative se match karte hain; koi bhi aur polynomial kisi jagah kisi slope ya curvature se disagree karega.

Edge cases

EC1. ke liye series method mein kya hota hai agar tum ko sirf order tak expand karo?
Tumhe milta hai, ek apparent . Tumhe ko order tak expand karna hoga taaki bachne wala sahi limit de.
EC2. Kya (centre) ek Maclaurin approximation ke liye special case hai?
Haan — par error term hota hai, isliye bhi exact hai: polynomial zero remainder ke saath perfectly reproduce karta hai.
EC3. series apne left endpoint par kya karti hai?
Yeh ban jaati hai, negative harmonic series, jo tak diverge karti hai — se match karta hai. Isliye interval se exclude hai.
EC4. Agar degree ka ek polynomial hai, toh ke baare mein uski Taylor series kya hai?
Polynomial khud — order se aage saare derivatives zero hain, isliye aur "series" terminate ho jaati hai; approximation har jagah exact hai.
EC5. Kya do functions same first Taylor terms share kar sakte hain phir bhi alag ho sakte hain?
Haan — e.g. aur (jahan ) ke identical zero Maclaurin coefficients hain lekin ke liye alag hain; finitely many terms match karna kabhi equality guarantee nahi karta.
EC6. Series se kya hai, aur yeh trouble kyun nahi hai?
; numerator mein denominator ko cleanly cancel karta hai, constant term bachta hai.
EC7. Ek alternating series ke liye, "error first omitted term" rule kab fail hota hai?
Jab terms size mein monotonically decreasing nahi hote; bound require karta hai ki har term pichhle se chhota ho. Agar ye grow karein, toh Lagrange bound use karo.
Recall Traps ka one-line summary

Kaafi terms rakho (denominator ka order), radius ka respect karo, trust karne se pehle bound karo, powers match karke cancel karo, aur kabhi "exact remainder form" aur "ek usable number" ko confuse mat karo.

Connections