Ratio & Root Test | Shot Revision & Question Practice | CSIR NET Maths 2026 | Lec-25 | IFAS скачать в хорошем качестве

Ratio & Root Test | Shot Revision & Question Practice | CSIR NET Maths 2026 | Lec-25 | IFAS

In this lecture by IFAS, Manoj Sir provides a targeted revision of series of real numbers, focusing on essential tests for convergence. This session is part of the ongoing real analysis series designed for CSIR NET Mathematical Science 2026 and GATE Mathematics aspirants, aiming to strengthen problem-solving speed through practical examples and direct applications of series tests. 🔥 Join India's Best Mathematics Coaching for CSIR NET and GATE Exams Live Online Course & Recorded Course - https://ifasonline.com/ 👉🏻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: Difference between Convergent, Divergent, and Oscillatory Sequences Behavior of $(-1)^n$ and $\sin(n)$ Bounded vs. Unbounded Sequences Range sets and limit points of $\sin(n)$ and $\sin^2(n)$ Analysis of $\sum \frac{\sin(n)}{n^2}$ and $\sum \frac{1}{n^2 \sin^2(n)}$ Ratio Test: Definition and Limit Evaluation Root Test: Criteria and Limit Superior Comparing Ratio and Root Tests Relationship between Limit Inferior/Superior of Ratio and Root Tests Radius of Convergence (ROC) for Power Series ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬ ⏱️ Timestamps - Jump to Your Topic: [03:27] Introduction and Welcome [05:48] Solving Previous Homework: Limit of $n^{\ln(n)/n}$ [08:46] Discussion on $(-1)^n$: Convergent vs. Oscillatory Sequences [11:34] Deep Dive: Convergence of $\sum \frac{1}{n^2 \sin^2(n)}$ [19:53] Example: Ratio Test for $\sum \frac{3^n}{n}$ [23:31] Relationship between Ratio Test and Root Test Limits [25:47] Question Practice: $\sum \frac{(n!)^2}{(2n)!}$ [27:07] Root Test Application for alternating powers [30:33] Practice: Convergence of $\sum \frac{5^n}{n!}$ [32:08] Solving $1^\infty$ forms using Root Test [35:08] Mentorship: CSIR NET 2026 Batch Details and Features [44:10] Advanced Problem: Dealing with Inconclusive Ratio Tests ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬ Lecture Summary: In this session, Manoj Sir focuses on Ratio & Root Tests, providing a bridge between conceptual theory and competitive exam applications. The lecture covers everything from the basic definitions of convergence to advanced problem-solving techniques for complex functional forms involving factorials and trigonometric terms. Manoj Sir also resolves doubts regarding the classification of sequences and explains why the root test is often a more powerful tool for specific power-based series. The session concludes with a look at Raabe's Test for cases where the standard ratio test fails to provide a conclusion. ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬ 🔗 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