AP Calculus BC Practice Questions

The series converges for all real values of x. It is important to note that x can take on any real number, and the series will still converge.

The series converges for all real values of x except for x = 0. This is because the series becomes undefined at x = 0 and therefore cannot be evaluated.

The limit comparison test can be used to determine if a series converges or diverges by comparing it to another series that is known to converge or diverge. The Limit Comparison Test: If $ ewline$ $ ewline$ lim$_{n→∞}$ [a(n)/b(n)] = c, where c is a finite number and c > 0, then both series either converge or both diverge.

The series diverges. This is because the harmonic series is a known divergent series.

The limit comparison test can be used to determine convergence or divergence of a series. If the limit of the ratio of the terms of two series is a positive finite number, then the two series either both converge or both diverge.

The series diverges for all values of x where the absolute value of x is greater than 3. This is because the series does not converge if the absolute value of x is greater than 3.

For the series $ ewline$ $ ewline$ ∑(1/n^p), where p is a real number, it converges if p is greater than 1; it diverges if p is less than or equal to 1. Consider the cases p = 1 and p < 1. If p < 1, then the given series becomes a harmonic series which diverges.

If f(x) is positive, continuous, and decreasing for x greater than or equal to 1, and if the series ∑f(n) converges, then the limit of f(x) as x approaches infinity must be zero. The integral test states that the series ∑f(n) converges if and only if the corresponding improper integral ∫f(x) dx converges.

If g(x) is positive, continuous, and decreasing for x greater than or equal to 1, then the integral test can be used to determine convergence. This means that the first derivative, dg/dx, must be negative.

The integral test can be used to determine if a series converges or diverges. If the improper integral of the function corresponding to the series diverges, then the series diverges. The integral test states that the series ∑f(n) converges if and only if the corresponding improper integral ∫f(x) dx converges.

If h(n) does not have a finite limit as n approaches infinity, then the integral test cannot be used to determine convergence or divergence. However, since there is no finite limit, the series diverges. Additionally, the terms of the series must have a limit of 0 for the series to converge. If the terms do not approach 0, then the series diverges.

If f(x) is positive, continuous, and decreasing for x greater than or equal to 1, and ∑f(n) converges to a finite value k, then this value k represents the sum of the infinite series. The convergence of the series means that the sum of the infinite series approaches a finite value.

If the series converges for all values of n, then the series is a convergent series. This means that the terms of the series approach a finite value as n approaches infinity. In other words, the sum of the series exists and is finite. Such behavior is characteristic of convergent series and ensures the series does not diverge.

The sum of the series is the limit of the nth partial sum as n approaches infinity. This is the sum the series converges to. The nth partial sum is the sum of the first n terms of the series. In other words, the sum of the infinite series is the limit of the sum of the first n terms as n becomes infinitely large. This limit represents the value the series approaches as we include more and more terms, and it defines the overall sum of the series.

If the sequence a_n diverges, then the infinite series also diverges. Divergence of the sequence indicates that the terms of the sequence do not approach a finite number as n goes to infinity. Consequently, the series formed by those terms also diverges as the sum of these terms will not converge to a finite value.

If p is greater than 1 for the term 1/n^p, then the infinite series converges. The series converges because the value of n^p grows much faster than n as n approaches infinity, indicating that the individual terms decrease rapidly enough for the series to converge to a finite sum. This is a fundamental result for understanding the convergence of p-series.

The series diverges for all cases. This is because the series does not converge. The provided information indicates that the listed series do not converge, meaning their sums do not approach a finite value. These series are divergent.

The term test can be used to determine divergence for series I and III. The nth term test for divergence states that if the limit of the nth term of a series does not equal zero, then the series diverges. If the limit does not equal zero, then the series does not approach zero as n approaches infinity. Therefore, the condition for the series to converge, where the terms must eventually become arbitrarily small, is not satisfied. Therefore, the series diverges.