Serie numeriche da esame

Serie esame

∑ n! ₃ⁿ / nʳ − 2 ∑ (n² + 7)ⁿ / (2n² + 1) − 3 ∑ 4ⁿ / 5ⁿ − 1 / n+ ∑ 2ⁿ / eⁿ+1 − 5 ∑ eⁿ / 2ⁿ − 6 ∑ cos²n / n² + 2

Serie esame

∑ n! ₃ⁿ / nⁿ - 2 ∑ (n² + 7) / (2n² + 1) - 3 ∑ 4ⁿ / 5ⁿ - 1 / n + ∑ 2ⁿ / eⁿ⁺¹ - 5 ∑ eⁿ / 2ⁿ - 6 ∑ cos²n / n² + 2

Serie esame

∑ _m=1 ^∞ ½^m / m^m - 2 ∑ _m=1 ^∞ (m³ + 7) / (2m+1)^m - 3 ∑ _m=1 ^∞ 4^m / 5^m - 1/m

∑ _m=1 ^∞ 2/ e^m+1 - 5 ∑ _m=1 ^∞ e^m / 2^m - 6 ∑ _m=1 ^∞ cos²m / m² + 2

∑ _m=1 ^∞ 1/r⁴ + √m / 6^m + ln(m+1) - 8 ∑ _m=1 ^∞ sen²m / 3^m - 9 ∑ _m=1 ^∞ (ln2)^m / 2m + 3

∑ _m=1 ^∞ sen²(2/m) - 11 ∑ _m=1 ^∞ cos^mn / m³ - 12 ∑ _m=1 ^∞ (m+1) / 2m³ - 3m + 2

Σ∑ _m=1 ^∞ m! √m / m^m

Verifica cond. necessaria di convergenza

lim _n→∞ m! √m / m^m = 0 m^1/3 m^-m ~ m! / m^m

lim _n→∞ (m+1) / m^∞m / x!

lim _n→∞ ln (m+1)^m = ln

HOPITAL f' = 1/(m) + 1/m1/m² + 2m + m = (m+1) / (m+1)

lim _m→∞ (m³ + m²)→ lim _m→∞ m³ + 2m²m-1/m²

lim _m→∞ (m + 1) / m³ + 2m + m= -1-lim m⁰ ln (¹/ε ) ≅ 0→ lim ^-1 1/l < 1

Converge

① ∑ ^{(n2 + 7)}/_{(2n² + 1)}ⁿ CONVERGE

lim_n→∞ √ⁿ (n² + 7)/2n² = lim_n→∞ (n² + 7)/2n² > lim_n→∞n⁶ (1 + ⁷/_n²) / n⁶ (2 + ⁷/_n²) = ¹/₂ < 1

Converge

③ ∑ ⁴/₅ⁿ CONVERGE

lim_n→∞ √^{n 1}/_5ⁿ = lim_n→∞ ⁴/₅^1/n = ⁴/₅ < 1

Converge

④ ∑ ²ⁿ/_eⁿ⁺¹ ~ ²ⁿ/_e⁴ CONVERGE

lim_n→∞ √^{n 2n}/_eⁿ = ²/_e < 1

Diverge

⑤ ∑ ^en/_2ⁿ lim_n→∞ √^{n en}/_2ⁿ = ^e/₂ > 1 DIVERGE

Converge

⑥ ∑ ^cos2n/_{n² + 2} ~ ^cos2n/_n² CONVERGE

0,5 cos2n/n2 < 1/n2 1/n2 CONVERGE ∑1/n2 α > 1 CONVERGE

Converge

⑦ ∑^{n4 + √n}/_{6ⁿ + ln(n + 1)} ~ ⁿ⁴/_6ⁿ a_n+1 = ^{(n + 1)4}/_6ⁿ6 CONVERGE

lim_n→∞ (n + 1)⁴ / 6ⁿ6 = lim_n→∞ ((n + 1)⁴ / 6ⁿ4) HOPITAL

lim_n→∞ ^{(n + 1)3}/_2/4n³ HOPITAL lim_n→∞ ^{6(n + 1)2}/_6²n² -> HOPITAL lim_n→∞ ^{2n + 2}/_12n = lim_n→∞^{n(2 + 2/_n) - 1} 6/12ⁿ²