Real Analysis - Sequences & Series

Q1: Which of the following is TRUE about a convergent sequence in R?

It is bounded It is monotone It has a unique limit Both A and C

Q2: What is the sum of the series Σ_{n=1}∞ 1/n²?

π²/6 π/6 π²/4 π/3

Q3: Which of the following tests can determine the convergence of the series Σ_{n=1}∞ 1/n!?

Ratio test Root test Comparison test All of the above

Q4: For a sequence {a_n}, what does it mean for the series Σ a_n to converge absolutely?

Σ |a_n| converges Σ a_n converges conditionally |a_n| → 0 Σ a_n diverges

Q5: What is the radius of convergence of the power series Σ_{n=0}∞ x^n/n!?

∞ 1 0 1/e

Q6: Which of the following is TRUE about uniformly convergent sequences of functions?

Uniform convergence implies pointwise convergence Pointwise convergence implies uniform convergence Uniform convergence preserves continuity Both A and C

Q7: What is the limit of the sequence a_n = (1 + 1/n)^n?

e 1 ∞ 0

Q8: For the series Σ a_n, if lim |a_{n+1}/a_n| = L, what does the ratio test state?

If L < 1, the series converges absolutely If L > 1, the series diverges If L = 1, the test is inconclusive All of the above

Q9: Which of the following sequences is NOT Cauchy?

a_n = 1/n a_n = (-1)^n/n a_n = n a_n = 1/n²

Q10: What is the sum of the geometric series Σ_{n=0}∞ r^n for |r| < 1?

1/(1-r) 1/(1+r) 1/(r-1) 1-r

Q11: Which of the following statements about the Weierstrass M-test is TRUE?

It gives a sufficient condition for uniform convergence of a series of functions It gives a necessary condition for uniform convergence It can only be used for power series It applies only to sequences of functions

Q12: For a sequence {a_n}, what is the relationship between convergence and boundedness?

Every convergent sequence is bounded Every bounded sequence is convergent Convergent sequences are always unbounded There is no relationship

Q13: What is the limit of the sequence a_n = sin(n)/n?

0 1 ∞ Does not exist

Q14: Which of the following series converges conditionally?

Σ_{n=1}∞ (-1)^{n+1}/n Σ_{n=1}∞ 1/n Σ_{n=1}∞ 1/n² Σ_{n=1}∞ 1/n!

Q15: What is the Cauchy condensation test used for?

Determining convergence of series with non-increasing positive terms Testing absolute convergence Finding the sum of a series Testing uniform convergence