PYQ on Series of Functions | Real Analysis | CSIR NET Mathematical Science | IFAS
In this lecture by IFAS, Manoj Sir focuses on practical problem-solving for Series and Sequences of Functions, a critical topic for CSIR NET Mathematical Science and GATE aspirants. The session covers key convergence tests, the Dini's Theorem, and the exchange of limits with integrals, providing students with the tools to tackle complex Real Analysis problems in competitive exams.
🔥 Join India's Best Mathematics Coaching for CSIR NET and GATE Exams - IFAS Online! STARTING SOON: Super Emotion Batch 7.0 (From 12th Jan 2026)
👉🏻IFAS CSIR UGC NET Mathematical Science Result👈🏻:
CSIR - NET June 2025 - Total Selections: 4200+ (CSIR UGC JRF AIR 1, 2, 6, 6, 10.....)
CSIR - NET Dec 2024 - Total Selections: 4000+ (CSIR UGC JRF AIR 5, 6, 6, 7, 8, 9, 10......)
CSIR - NET June 2024 - Total Selections: 2000+ (CSIR UCG JRF AIR 3, 10, 13, 15, 16.....)
CSIR - NET Dec - 2023: Total Selections: 1200+ (CSIR UGC JRF AIR 2, 4, 5, 7, 12, 15, 18, 19......)
CSIR - NET June - 2023: Total Selections: 1000+ (CSIR UGC JRF AIR 2, 6, 8, 9.....)
▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬
🎯 Target Audience: This video is highly beneficial for students preparing for:
CSIR NET Mathematics, GATE Mathematics, MH SET Mathematics, BARC, NBHM
▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬
👇 Watch the Full Lecture to Master These Topics:
Pointwise Convergence
Uniform Convergence (UC)
Dini's Theorem
Integration and Limits
Derivative Convergence
Weierstrass M-Test
▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬
⏱️ Timestamps - Jump to Your Topic:
[00:55] Introduction and Exam Updates (CSIR NET Dec 2025)
[06:44] Example 1: Analyzing $f_n(x) = \frac{1}{1+n^2x^2}$ (Pointwise vs. Uniform)
[13:21] Dini’s Theorem Application and Integration Limit Check
[16:33] Example 2: Analysis of $f_n(x) = \frac{1}{2nx+1}$ on $(0, 1)$
[20:00] Limit Function Evaluation and $M_n$-Test for Non-Uniform Convergence
[27:57] Example 3: $f_n(x) = \frac{e^{-x/n}}{n}$ on $[0, \infty)$
[31:44] Uniform Convergence of Derivatives and Integral Comparison
[34:55] Series of Functions: Uniform Convergence using Weierstrass M-Test
[39:05] Final Example: Convergence of $\sum \frac{e^{-nx}}{2n^2}$ and Session Wrap-up
▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬
Lecture Summary:
The session is dedicated to Series and Sequences of Functions PYQs, bridging the gap between theoretical Real Analysis and exam-style problems. Manoj Sir demonstrates how to calculate limit functions, apply the $M_n$-test, and use Dini's Theorem for uniform convergence. A significant portion of the lecture is spent verifying whether integration and limit operations can be interchanged, even when uniform convergence holds on non-compact domains. The session concludes with a preview of upcoming PSC sessions and further preparation tips for the CSIR NET December cycle.
▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬
🔗 Important Links:
🌍 Visit our Website: https://ifasonline.com
📞 Contact for Queries: 9172266888
📲 WhatsApp: https: https://whatsapp.com/channel/0029VbCV...
🔥 Qualification Prediction (Free Tool): https://ifasonline.com/QualificationP...
📘 Get Detailed Free Counselling: https://ifasonline.com/csir-net/mathe...
▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬
Download IFAS: 👇
📲Android: https://play.google.com/store/apps/de...
📲 Ios: https://apps.apple.com/us/app/ifas/id...
▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬
CSIR NET Mathematical Science Live & Recorded Course –
https://ifasonline.com/csir-net/mathe...
GATE Mathematical Science Recorded Course –
https://ifasonline.com/gate/mathemati...
MH SET Mathematical Science Recorded Course –
https://ifasonline.com/mh-set/mathema....
📘 Buy CSIR NET Mathematical Science Books:
https://ifasonline.com/csir-net/mathe...
▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬
📞 Contact for Admission: 9172266888
Stay Updated Via Other Social Media:
✔Telegram: https://t.me/NET_GATE_SET_Maths_Discu...
✔Instagram: / csir_net_mathematical_science
✔Facebook: / ifasmaths
✔WhatsApp: https: https://whatsapp.com/channel/0029VbCV...
▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬
#csirnetjune2026 #csirnetmaths #csirnetmathematics #csirnetmathematicalscience #csirnetpreparation #ifas