Definite Integral as Limit of a Sum

JEE Main Mathematics · Definite Integration · 6 questions, latest first

• Let [·] denote the greatest integer function and f(x)= lim_n → ∞ 1/(n^3) Σ_k=1^n[(k^2)/3^x]. Then 12 Σ_j=1^∞ f(j) is equal to…202621 January, Shift 2 · Q24
• Let lim_n → ∞(n/(√(n^4+1))-(2 n)/((n^2+1) √(n^4+1))+n/(√(n^4+16))-(8 n)/((n^2+4) √(n^4+16))..+…+n/(√(n^4+n^4))-(2 n ·…20249 April, Shift 1 · Q28
• The value of lim_n → ∞ Σ_k=1^n n^3/((n^2+k^2)(n^2+3 k^2)) is:202430 January, Shift 1 · Q10
• Among (S1): lim_n → ∞ 1/n^2(2+4+6+… …+2 n)=1 (S2): lim_n → ∞ 1/(n^16)(1^15+2^15+3^15+… …+n^15)=1/16202313 April, Shift 1 · Q7
• lim_n → ∞[1/(1+n)+1/(2+n)+1/(3+n)+…+1/(2 n)] is equal to20231 February, Shift 1 · Q70
• lim_n → ∞ 3/n{4+(2+1/n)^2+(2+2/n)^2+…+(3-1/n)^2} is equal to202330 January, Shift 2 · Q70