MATH1005 test 4
Key concepts in relation to sequences and infinite series
Sequences: An ordered list of numbers that may converge to a limit or diverge to infinity. For example, the sequence defined by a_n = 1/n converges to 0 as n approaches infinity.
Squeeze theorem: A principle that states if a function is squeezed between two other functions that both converge to the same limit, then the squeezed function also converges to that limit. For example, if f(x) ≤ g(x) ≤ h(x) for all x in some interval and lim as x approaches a of f(x) = lim as x approaches a of h(x) = L, then lim as x approaches a of g(x) = L.
Bounded above, bounded below, monotone: A sequence is bounded above if there exists a number greater than or equal to every term in the sequence, bounded below if there exists a number less than or equal to every term, and monotone if it is either non-increasing or non-decreasing. For instance, the sequence a_n = 1/n is bounded below by 0 and bounded above by 1.
Geometric series to find sum of series: A series in which each term is a constant multiple (common ratio) of the previous term; its sum can be calculated using the formula S = a/(1 - r), where S is the sum, a is the first term, and r is the common ratio (|r| < 1). For example, the series 1 + 1/2 + 1/4 + 1/8 + … is a geometric series where a = 1 and r = 1/2, converging to S = 1/(1 - 1/2) = 2.
Integral test: A method for determining the convergence of a series by comparing it to an improper integral; if the integral converges, so does the series. For example, to test the series Σ 1/n², one can compare it with the integral ∫ 1/x² dx from 1 to infinity, which converges to 1.
Approximations of series: Techniques used to estimate the sum of a series, often through partial sums or other means of calculation. For instance, the sum of the first k terms of a harmonic series can be approximated using the natural logarithm: H_n ≈ ln(n) + γ, where γ is the Euler-Mascheroni constant.
Comparison test: A method to determine the convergence of one series by comparing it to another known series; if the known series converges, so does the other if its terms are less than those of the known series. For example, if Σ an converges and 0 ≤ bn ≤ an for all n, then Σ bn also converges.
Limit comparison test: A refinement of the comparison test that establishes the convergence of two series based on the limit of their term ratios. For example, if lim (n→∞) (an/bn) = c where 0 < c < infinity, then both Σ an and Σ bn converge or diverge together.
Alternating series: A series in which the terms alternate in sign; often used to analyze convergence using the alternating series test. An example is the series Σ (-1)ⁿ/(n), which converges due to the alternating nature of its terms.
Alternating series test: A specific test to determine the convergence of an alternating series, requiring that the absolute values of the terms decrease monotonically to zero. For example, for the series Σ (-1)ⁿ/(n²), we check that |a_n| = 1/n² decreases and approaches 0, thus the series converges.
Approximation of alternating series: Techniques used to estimate the sum of an alternating series by considering the first few terms or the remainder. For the series Σ (-1)ⁿ/(n), the first few terms provide a reasonable approximation of the total sum.
Absolute and conditional convergence: A series is absolutely convergent if the series formed by taking the absolute values of its terms converges; a series is conditionally convergent if it converges, but the series of absolute values does not. An example is the series Σ (-1)ⁿ/(n), which is conditionally convergent, whereas Σ 1/n² is absolutely convergent.
Ratio test: A test that determines the convergence of a series by examining the limit of the ratio of consecutive terms; if the limit is less than 1, the series converges absolutely. For example, in the series Σ n!/n², applying the ratio test gives a result that confirms the series converges.
Root test: A test that assesses the convergence of a series by examining the limit of the nth root of the absolute value of its terms; similar to the ratio test, it provides convergence criteria based on the limit value. For example, the series Σ (1/n^n) can be evaluated using the root test, confirming it converges as its limit approaches 0.