Page 2 For more notes, call 8130648819 Since, logn is an increasing squence, u is a decreasing sequence u u n y Cauchy s condensation test,the series ? u and ? u converge or diverge together Now ? u ? log ? nlog log ? n Consider v n v n lim v lim n y Cauchy s root test,?v is divergent ? u is divergent ? u is divergent (ii) Here u nlogn Since, nlogn is an increasing sequence, u is a decreasing sequence i e u u n y Cauchy s condensation test,the series ? u and ? u converge or diverge together Now ? u ? log ? nlog log ? n Since,? n is divergent ? u is divergent ? u is divergent (iii) Here u nvlogn Since nvlogn is an increasing sequence, u is a decreasing sequence u u n y Cauchy s condensation test,the series ? u and ? u converge or diverge together Now ? u ? vlog ? vnlog vlog ? vn Since,? vn is divergent ( p ) ? u is divergent ? u is divergent (iv) Do yourself (v) Here u n(logn) Page 3 For more notes, call 8130648819 Since, logn is an increasing squence, u is a decreasing sequence u u n y Cauchy s condensation test,the series ? u and ? u converge or diverge together Now ? u ? log ? nlog log ? n Consider v n v n lim v lim n y Cauchy s root test,?v is divergent ? u is divergent ? u is divergent (ii) Here u nlogn Since, nlogn is an increasing sequence, u is a decreasing sequence i e u u n y Cauchy s condensation test,the series ? u and ? u converge or diverge together Now ? u ? log ? nlog log ? n Since,? n is divergent ? u is divergent ? u is divergent (iii) Here u nvlogn Since nvlogn is an increasing sequence, u is a decreasing sequence u u n y Cauchy s condensation test,the series ? u and ? u converge or diverge together Now ? u ? vlog ? vnlog vlog ? vn Since,? vn is divergent ( p ) ? u is divergent ? u is divergent (iv) Do yourself (v) Here u n(logn) For more notes, call 8130648819 Case I When p n(logn) n n Since,? n diverges,by comparison test ? n(logn) diverges Case II When p Since n(logn) is an increasing sequence, u is a decreasing sequence i e u u n y Cauchy s condensation test,the series ? u and ? u converge or diverge together Now ? u ? (log ) ? ( log ) (log ) ? n Since,? n is convergent if p and divergent if p ? u is convergent if p and divergent if p ? u is convergent if p and divergent if p Hence,? u is convergent if p and divergent if p (vi) Here u (logn) Case I When p ,u Since, lim u ,? diverges Case II when p ,let p q,where q Since, lim u lim (logn) lim (logn) ? u diverges Case III When p Since, (logn) is an increasing sequence, u is a decreasing sequence i e u u n y Cauchy s condensation test,the series ? u and ? u converge or diverge together Now ? u ? (log ) ? (nlog ) (log ) ? n Consider, v n so that v / (n / ) lim v lim (n ) y Cauchy s root test,?v is divergent ? u is divergent ? u is divergent Hence,? u is divergent for all values of p Page 4 For more notes, call 8130648819 Since, logn is an increasing squence, u is a decreasing sequence u u n y Cauchy s condensation test,the series ? u and ? u converge or diverge together Now ? u ? log ? nlog log ? n Consider v n v n lim v lim n y Cauchy s root test,?v is divergent ? u is divergent ? u is divergent (ii) Here u nlogn Since, nlogn is an increasing sequence, u is a decreasing sequence i e u u n y Cauchy s condensation test,the series ? u and ? u converge or diverge together Now ? u ? log ? nlog log ? n Since,? n is divergent ? u is divergent ? u is divergent (iii) Here u nvlogn Since nvlogn is an increasing sequence, u is a decreasing sequence u u n y Cauchy s condensation test,the series ? u and ? u converge or diverge together Now ? u ? vlog ? vnlog vlog ? vn Since,? vn is divergent ( p ) ? u is divergent ? u is divergent (iv) Do yourself (v) Here u n(logn) For more notes, call 8130648819 Case I When p n(logn) n n Since,? n diverges,by comparison test ? n(logn) diverges Case II When p Since n(logn) is an increasing sequence, u is a decreasing sequence i e u u n y Cauchy s condensation test,the series ? u and ? u converge or diverge together Now ? u ? (log ) ? ( log ) (log ) ? n Since,? n is convergent if p and divergent if p ? u is convergent if p and divergent if p ? u is convergent if p and divergent if p Hence,? u is convergent if p and divergent if p (vi) Here u (logn) Case I When p ,u Since, lim u ,? diverges Case II when p ,let p q,where q Since, lim u lim (logn) lim (logn) ? u diverges Case III When p Since, (logn) is an increasing sequence, u is a decreasing sequence i e u u n y Cauchy s condensation test,the series ? u and ? u converge or diverge together Now ? u ? (log ) ? (nlog ) (log ) ? n Consider, v n so that v / (n / ) lim v lim (n ) y Cauchy s root test,?v is divergent ? u is divergent ? u is divergent Hence,? u is divergent for all values of p For more notes, call 8130648819 Ex Using Cauchy s condensation test,discuss the convergence of ? logn n Solution Here u logn n n Consider f(x) logx x ,x f (x) x x logx x logx x f (x) logx logx x e ( loge ) f(x) is a decreasing function when x e u u ( e ) u is a decreasing sequence of positive terms y Cauchy s condensation test,the series ? u and ? u converge or diverge together Now, ? u ? log ? nlog log ? n Since,? n is divergent,? u is divergent ? u is divergent Ex Using Cauchy s condensation test,discuss the convergence of ? ( logn n ) Solution Here u ( logn n ) n consider f(x) ( logx x ) , x f (x) ( logx x ) ( logx x ) f (x) logx logx x e ( loge ) f(x) is decreasing function where x e u u n ( e ) u is a decreasing sequence of positive terms y Cauchy s condensation test,the series ? u and ? u converge or diverge together Now ? u ? 4 log 5 ? (nlog ) (log ) ? n Consider v n so that v y Cauchy s root test,? v converges ? u converges ? u converges Page 5 For more notes, call 8130648819 Since, logn is an increasing squence, u is a decreasing sequence u u n y Cauchy s condensation test,the series ? u and ? u converge or diverge together Now ? u ? log ? nlog log ? n Consider v n v n lim v lim n y Cauchy s root test,?v is divergent ? u is divergent ? u is divergent (ii) Here u nlogn Since, nlogn is an increasing sequence, u is a decreasing sequence i e u u n y Cauchy s condensation test,the series ? u and ? u converge or diverge together Now ? u ? log ? nlog log ? n Since,? n is divergent ? u is divergent ? u is divergent (iii) Here u nvlogn Since nvlogn is an increasing sequence, u is a decreasing sequence u u n y Cauchy s condensation test,the series ? u and ? u converge or diverge together Now ? u ? vlog ? vnlog vlog ? vn Since,? vn is divergent ( p ) ? u is divergent ? u is divergent (iv) Do yourself (v) Here u n(logn) For more notes, call 8130648819 Case I When p n(logn) n n Since,? n diverges,by comparison test ? n(logn) diverges Case II When p Since n(logn) is an increasing sequence, u is a decreasing sequence i e u u n y Cauchy s condensation test,the series ? u and ? u converge or diverge together Now ? u ? (log ) ? ( log ) (log ) ? n Since,? n is convergent if p and divergent if p ? u is convergent if p and divergent if p ? u is convergent if p and divergent if p Hence,? u is convergent if p and divergent if p (vi) Here u (logn) Case I When p ,u Since, lim u ,? diverges Case II when p ,let p q,where q Since, lim u lim (logn) lim (logn) ? u diverges Case III When p Since, (logn) is an increasing sequence, u is a decreasing sequence i e u u n y Cauchy s condensation test,the series ? u and ? u converge or diverge together Now ? u ? (log ) ? (nlog ) (log ) ? n Consider, v n so that v / (n / ) lim v lim (n ) y Cauchy s root test,?v is divergent ? u is divergent ? u is divergent Hence,? u is divergent for all values of p For more notes, call 8130648819 Ex Using Cauchy s condensation test,discuss the convergence of ? logn n Solution Here u logn n n Consider f(x) logx x ,x f (x) x x logx x logx x f (x) logx logx x e ( loge ) f(x) is a decreasing function when x e u u ( e ) u is a decreasing sequence of positive terms y Cauchy s condensation test,the series ? u and ? u converge or diverge together Now, ? u ? log ? nlog log ? n Since,? n is divergent,? u is divergent ? u is divergent Ex Using Cauchy s condensation test,discuss the convergence of ? ( logn n ) Solution Here u ( logn n ) n consider f(x) ( logx x ) , x f (x) ( logx x ) ( logx x ) f (x) logx logx x e ( loge ) f(x) is decreasing function where x e u u n ( e ) u is a decreasing sequence of positive terms y Cauchy s condensation test,the series ? u and ? u converge or diverge together Now ? u ? 4 log 5 ? (nlog ) (log ) ? n Consider v n so that v y Cauchy s root test,? v converges ? u converges ? u converges For more notes, call 8130648819 Ex Using Cauchy s condensation test,discuss the convergence of ? (nlogn) Solution Here u (nlogn) , n Case I When p ,u Since, lim u , ? u is divergent Case II When p ,let p q where q u (nlogn) (nlogn) u as n Since, lim u , u is divergent Case III When p , u is a decreasing sequence y Cauchy s condensation test,the series ? u and ? u converge or diverge together Now ? u ? ( log ) ? ( nlog ) ? ( ) n (log ) (log ) ? ( ) n If p ,then ( ) so that ( ) and ( ) n n ut the series ? n is convergent for p ? ( ) n is convegent ? u is convergent for p If p ,then ? u log ? n ut ? n is divergent ? u is divergent for p If p ,then p so that ( ) ,i e ( ) and ( ) and ( ) n n ut the series ? n is divergent for p Hence,? u is convergent for p and divergent for p Def A series with terms alternatively positive and negative is called an alternating series Thus the series u u u u where u n is an alternating series and is briefly written as ?( ) u Leibnitz s Test on Alternating Series Statement The alternating series ?( ) u u u u u (u n) converges if (i) u u n and (ii) lim u Proof Let S denote the nth partial sum of the series ?( ) u S u u u u u u u u u (u u ) (u u ) (u u ) u u ,(u u ) (u u ) (u u ) u - u , u u and u n- the sequence S is bounded above Also S S u uRead More

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