Sum of Series: JEE Questions & Solutions

Let a₁=1 and for n ≥ 1, aₙ₊₁ = 1/2aₙ + (n²-2n-1)/n²(n+1)². Then |Σ_(n=1)^∞ (aₙ - 2/n²)| is equal to:

The positive integer n, for which the solutions of the equation x(x+2) + (x+2)(x+4) + … + (x+2n-2)(x+2n) = 8n/3 are two consecutive even…

Let 729, 81, 9, 1, … be a sequence and Pₙ denote the product of the first n terms of this sequence. If 2 Σ_(n=1)⁴⁰ (Pₙ)^(1/n) = (3^α -…

Let S=1/25!+1/(3! 23!)+1/(5! 21!)+… up to 13 terms. If 13S=2^k/n!, k∈ℕ, then n+k is equal to

The value of (1/3+4/7) +(1/3²+1/3×4/7+4/7²) +(1/3³+1/3²×4/7+1/3×4/7²+4/7³) +… up to infinite terms is

The value of Σ_(k=1)^∞ (-1)^(k+1)(k(k+1)/k!) is:

Evaluate: 6/3²⁶+(10·1)/3²⁵+(10·2)/3²⁴+(10·2²)/3²³+…+(10·2²⁴)/3.

If Σ_(r=1)²⁵(r/(r⁴+r²+1))=p/q, where p and q are positive integers such that gcd(p,q)=1, then p+q is equal to.

If Σ_(r=1)ⁿ T_r = (2n-1)(2n+1)(2n+3)(2n+5)/64, then lim_(n → ∞) Σ_(r=1)ⁿ 1/T_r is equal to:

If 7 = 5 + 1/7(5 + α) + 1/7²(5 + 2α) + 1/7³(5 + 3α) + …, then the value of α is:

Let ⟨ aₙ ⟩ be a sequence such that a₀ = 0, a₁ = 1/2, and 2aₙ₊₂ = 5aₙ₊₁ - 3aₙ. n= 0,1,2,3.... Then Σ_(k=1)¹⁰⁰ a_k is equal to:

Let E₁: x²/9 + y²/4 = 1 be an ellipse. Ellipses Eᵢ are constructed such that their centres and eccentricities are the same as that of E₁,…

For positive integers n, if 4 aₙ = n² + 5n + 6 and Sₙ = Σ_(k=1)ⁿ ( 1/a_k), then the value of 507 S₂₀₂₅ is:

The value of lim_(n → ∞) ( Σ_(k=1)ⁿ (k³ + 6k² + 11k + 5)/(k + 3)!) is:

Let S = ℕ ∪ {0}. Define a relation R from S to ℝ by: R = { (x, y): logₑ y = x logₑ (2/5), x ∈ S, y ∈ ℝ }. Then, the sum of all the elements…

Let Pₙ = αⁿ + βⁿ, n ∈ ℕ. If P₁₀ = 123, P₉ = 76, P₈ = 47 and P₁ = 1, then the quadratic equation having roots 1/α and 1/β is:

If the sum of the first 10 terms of the series (4 · 1)/(1 + 4 · 1⁴) + (4 · 2)/(1 + 4 · 2⁴) + (4 · 3)/(1 + 4 · 3⁴) + … is m/n, where gcd(m,…

The sum 1 + 3 + 11 + 25 + 45 + 71 + … upto 20 terms, is equal to

The sum 1 + (1 + 3)/2! + (1 + 3 + 5)/3! + (1 + 3 + 5 + 7)/4! + … upto ∞ terms, is equal to

1 + 3 + 5² + 7 + 9² + … upto 40 terms is equal to

The sum of the infinite series cot⁻¹ ( 7/4) + cot⁻¹ ( 19/4) + cot⁻¹ ( 39/4) + cot⁻¹ ( 67/4) + … is:

If the sum of the first 20 terms of the series (4· 1)/(4 + 3· 1² + 1⁴) + (4· 2)/(4 + 3· 2² + 2⁴) + (4· 3)/(4 + 3· 3² + 3⁴) + (4· 4)/(4 + 3·…

If α is a root of the equation x² + x + 1 = 0 and Σ_(k=1)ⁿ ( α^k + 1/α^k)² = 20, then n is equal to

If 1/1⁴ + 1/2⁴ + 1/3⁴ + … ∞ = π⁴/90, 1/1⁴ + 1/3⁴ + 1/5⁴ + … ∞ = α, 1/2⁴ + 1/4⁴ + 1/6⁴ + … ∞ = β, then α/β is equal to:

If 8 = 3 + 1/4(3 + p) + 1/4²(3 + 2p) + 1/4³(3 + 3p) + …, then the value of p is:

If α and β are the roots of x² - x - 1 = 0, and Sₙ = 2023αⁿ + 2024βⁿ, then:

Let α = 1² + 4² + 8² + 13² + 19² + 26² +... up to 10 terms and β = Σ_(n=1)¹⁰ n⁴. If 4α - β = 55k + 40, then k is equal to:

The sum of the series: ( 1/(1 - 3·(1²) + 1⁴)) + ( 2/(1 - 3·(2²) + 2⁴)) + ( 3/(1 - 3·(3²) + 3⁴)) + … up to 10 terms

The value of (1 × 2² + 2 × 3² + … + 100 × (101)²)/(1² × 2 + 2² × 3 + … + 100² × 101) is:

Let ABC be an equilateral triangle. A new triangle is formed by joining the midpoints of all sides of the triangle ABC, and the same…

If (1/(α+1) + 1/(α+2) + … + 1/(α+1012)) - (1/(2·1) + 1/(4·3) + 1/(6·5) + … + 1/(2024·2023)) = 1/2024, then α is equal to:

Let f(x) be a function such that f(x + y) = f(x)f(y) for all x, y ∈ ℕ. If f(1) = 3 and Σ_(k=1)ⁿ f(k) = 3279, then the value of n is:

If (1³ + 2³ + 3³ + … (up to n terms))/(1 · 3 + 2 · 5 + 3 · 7 + … (up to n terms)) = 9/5, then the value of n is:

Consider a function f: ℕ → ℝ satisfying f(1) + 2f(2) + 3f(3) + … + x f(x) = x(x + 1) f(x) for x ≥ 2, with f(1) = 1. Then 1/f(2022) +…

Let a₁ = b₁ = 1 and aₙ = aₙ₋₁ + (n-1), bₙ = bₙ₋₁ + aₙ₋₁, ∀ n ≥ 2. If S = Σ_(n=1)¹⁰ bₙ/2ⁿ and T = Σ_(n=1)⁸ n/2ⁿ⁻¹, then 2⁷ (2S - T) is equal…

If aₙ = (-2)/(4n² - 16n + 15), then a₁ + a₂ + … + a₂₅ is equal to:

Let Σ_(n=0)^∞ (n³ ( (2n)!) + (2n-1)(n!))/(n!)(2n)! = a e + b/e + c, where a, b, c ∈ ℤ and e = Σ_(n=0)^∞ 1/n!. Then a² - b + c is equal to.

Let S be the set of all values of a₁ for which the mean deviation about the mean of 100 consecutive positive integers a₁, a₂, a₃, …, a₁₀₀…

The sum 1² - 2 · 3² + 3.5² - 4.7² + 5.9² - … + 15.29², is:

The sum to 10 terms of the series 1/(1+1²+1⁴) + 2/(1+2²+2⁴) + 3/(1+3²+3⁴) + … is:

The sum Σ_(n=1)^∞ (2n² + 3n + 4)/(2n)! is equal to:

The sum of the first 20 terms of the series 5 + 11 + 19 + 29 + 41 + … is:

If gcd(m, n) = 1 and 1² - 2² + 3² - 4² + … + (2021)² - (2022)² + (2023)² = 1012 m² n then m² - n² is equal to:

If (20)¹⁹ + 2(21)(20)¹⁸ + 3(21)² (20)¹⁷ + … + 20(21)¹⁹ = k(20)¹⁹, then k is equal to:

Let S_K = (1 + 2 + … + K)/K and Σ_(j=1)ⁿ S_j² = n/A(Bn² + Cn + D), where A, B, C, D ∈ ℕ and A has the least value. Then:

If aₐ is the greatest term in the sequence aₙ = n³/(n⁴ + 147), n = 1, 2, 3, …, then a is equal to:

If Sₙ = 4 + 11 + 21 + 34 + 50 + … to n terms, then 1/60(S₂₉ - S₉) is equal to:

For k ∈ ℕ, if the sum of the series 1 + 4/k + 8/k² + 13/k³ + 19/k⁴ + … is 10, then the value of k is.

Let ⟨ aₙ ⟩ be a sequence such that a₁ + a₂ + … + aₙ = (n² + 3n)/(n+1)(n+2). If 28Σ_(k=1)¹⁰ 1/a_k = p₁p₂p₃ … p_m where p₁, p₂, …, p_m are…

The sum to 20 terms of the series 2· 2² - 3² + 2· 4² - 5² + 2· 6² - … is equal to

Let f(x) = Σ_(k=1)¹⁰ k x^k, x ∈ ℝ. If 2f(2) + f'(2) = 119(2)ⁿ + 1, then n is equal to.

Let [α] denote the greatest integer ≤ α. Then [√1] + [√2] + [√3] + … + [√120] is equal to.

If the sum of the series (1/2-1/3) +(1/2²-1/(2·3)+1/3²) +(1/2³-1/(2²·3)+1/(2·3²)-1/3³) +(1/2⁴-1/(2³·3)+1/(2²·3²)-1/(2·3³)+1/3⁴) +… is α/β,…