est the convergence of theseries 1 + 𝑥2 + 𝑥25 + 𝑥310 + ⋯

est the convergence of theseries 1 + 𝑥2 + 𝑥25 + 𝑥310 + ⋯

Describe Convergence of IT and OT.

Exercise n°1Identify the pattern of the following sequence.1, 1, 2, 3, 5, 8, 13, . ..Exercise n°2Write out the first four items of the sequences whose general terms are:a. 𝑎𝑎 𝑛𝑛 = 2𝑛𝑛 + 1b. 𝑎𝑎 𝑛𝑛 = 3𝑛𝑛+1c. 𝑎𝑎 𝑛𝑛 = 𝑛𝑛+1𝑛𝑛d. 𝑎𝑎 𝑛𝑛 = 1𝑛𝑛 2Exercise n°3What is the limit of the following sequences, and determine whether they converge or diverge?a. 𝑎𝑎 𝑛𝑛 = 2𝑛𝑛𝑛𝑛+1b. 𝑎𝑎 𝑛𝑛 = 3 + (−1)𝑛𝑛c. 𝑎𝑎 𝑛𝑛 = 2𝑛𝑛2𝑛𝑛 −1d. 𝑎𝑎 𝑛𝑛 = (−1)𝑛𝑛𝑛𝑛!e. 𝑎𝑎 𝑛𝑛 = 5𝑛𝑛+73𝑛𝑛−5f. 𝑎𝑎 𝑛𝑛 = 𝑛𝑛 2 +12𝑛𝑛−3Exercise n°4Identify the type of the following sequences.a. 𝑎𝑎 𝑛𝑛 = 2𝑛𝑛 + 3b. 𝑎𝑎 𝑛𝑛 = 2𝑛𝑛Exercise n°5Generate the general term of the following sequence.− 11 , 32 , − 76 , 1524 , − 31120 , …Exercise n°6Find the general term of the following sequence.Given the function 𝑓𝑓(𝑥𝑥) = 𝑒𝑒 𝑥𝑥3 𝑎𝑎 𝑛𝑛 = 𝑓𝑓(𝑛𝑛−1) (0)Exercise n°7Applying l’Hôpital’s rule, evaluate lim𝑛𝑛→∞(𝑛𝑛+1)𝑒𝑒 𝑛𝑛Exercise n°8Applying the squeeze theorem, determine the convergence or divergence of the following sequences:a. 𝑎𝑎 𝑛𝑛 = sin 𝑛𝑛𝑛𝑛 2b. 𝑎𝑎 𝑛𝑛 = (−1)𝑛𝑛 ∙ 1𝑛𝑛c. 𝑎𝑎 𝑛𝑛 = 𝑛𝑛!𝑛𝑛 𝑛𝑛Exercise n°9Investigate whether the following sequences are increasing, decreasing or neither.a. 𝑎𝑎 𝑛𝑛 = 𝑛𝑛𝑛𝑛+1b. 𝑎𝑎 𝑛𝑛 = 𝑛𝑛!𝑒𝑒 𝑛𝑛 , for 𝑛𝑛 ≥ 2Exercise n°10Show that the following sequence is bounded.𝑎𝑎 𝑛𝑛 = 3 − 4𝑛𝑛 2𝑛𝑛 2 + 1

area of convergence

Per quali 𝑥 reali è verificata la disequazione √𝑥2 − 1 > 2𝑥 ?A. 𝑥 ≥ −1 ☛ pagina 107B. 𝑥 ≤ −1 ☛ pagina 125C. −1 < 𝑥 < 1 ☛ pagina 93D. per nessun 𝑥 reale ☛ pagina 124E. 𝑥 ≥ 1