Harmonic Series and p-Series - AP Calculus BC

1. What is the condition for the convergence of a harmonic series?

   Always diverges. The harmonic series is famously always divergent.

2. Identify the convergence of $\sum_{n=1}^{\infty} \frac{1}{n^2}$.

   Converges. Since $p = 2 > 1$, this p-series converges.

3. What type of series is $\sum_{n=1}^{\infty} \frac{1}{n^{1.5}}$?

   p-series. Has the form $\sum \frac{1}{n^p}$ with $p = 1.5$.

4. What is the nature of $\sum_{n=1}^{\infty} \frac{1}{n^{2.1}}$?

   Convergent. Since $p = 2.1 > 1$, this p-series converges.

5. Does the series $\sum_{n=1}^{\infty} \frac{1}{n}$ converge or diverge?

   Diverges. The harmonic series is the classic divergent series.

6. Identify the series that converges: $\sum_{n=1}^{\infty} \frac{1}{n^3}$ or $\sum_{n=1}^{\infty} \frac{1}{n}$.

   $\sum_{n=1}^{\infty} \frac{1}{n^3}$. Since $p = 3 > 1$, the first series converges.

7. Does the series $\sum_{n=1}^{\infty} \frac{1}{n^{3.5}}$ converge?

   Converges. Since $p = 3.5 > 1$, this p-series converges.

8. Identify if the series $\sum_{n=1}^{\infty} \frac{1}{n^{0.6}}$ is convergent.

   Diverges. Since $p = 0.6 < 1$, this p-series diverges.

9. What condition makes a p-series converge?

   Converges if $p > 1$. When $p > 1$, the terms decrease fast enough for convergence.

10. What is the second term of a harmonic series?

    $\frac{1}{2}$. The second term in the harmonic series is $\frac{1}{2}$.

11. Does $\sum_{n=1}^{\infty} \frac{1}{n^{0.7}}$ converge or diverge?

    Diverges. Since $p = 0.7 < 1$, this p-series diverges.

12. Identify if $\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}$ converges or diverges.

    Converges. Since $p = 1.5 > 1$, this p-series converges.

13. Identify if the series $\sum_{n=1}^{\infty} \frac{1}{n^{1.5}}$ converges.

    Converges. Since $p = 1.5 > 1$, this p-series converges.

14. Which series is divergent: $\sum_{n=1}^{\infty} \frac{1}{n}$ or $\sum_{n=1}^{\infty} \frac{1}{n^2}$?

    $\sum_{n=1}^{\infty} \frac{1}{n}$. The harmonic series diverges while $\sum \frac{1}{n^2}$ converges.

15. Identify the type of series: $\sum_{n=1}^{\infty} \frac{1}{n^{0.9}}$.

    p-series. Has form $\sum \frac{1}{n^p}$ with $p = 0.9$.

16. What condition makes a p-series diverge?

    Diverges if $p \leq 1$. When $p \leq 1$, terms don't decrease fast enough.

17. What is the fourth term of a harmonic series?

    $\frac{1}{4}$. The fourth term in the harmonic series is $\frac{1}{4}$.

18. What is the definition of a harmonic series?

    The series $\sum_{n=1}^{\infty} \frac{1}{n}$. The classic divergent series with terms $\frac{1}{n}$.

19. What is the fifth term of a harmonic series?

    $\frac{1}{5}$. The fifth term in the harmonic series is $\frac{1}{5}$.

20. For which values of $p$ does $\sum_{n=1}^{\infty} \frac{1}{n^p}$ diverge?

    $p \leq 1$. P-series diverge when $p$ is at most 1.

21. State the formula for a p-series.

    $\sum_{n=1}^{\infty} \frac{1}{n^p}$. General form where $p$ determines convergence behavior.

22. For which $p$ is the series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ divergent?

    $p \leq 1$. P-series diverge when $p$ is at most 1.

23. State whether $\sum_{n=1}^{\infty} \frac{1}{n^{2.5}}$ converges.

    Converges. Since $p = 2.5 > 1$, this p-series converges.

24. Is $\sum_{n=1}^{\infty} \frac{1}{n^{0.5}}$ convergent or divergent?

    Divergent. Since $p = 0.5 \leq 1$, this p-series diverges.

25. What type of series is $\sum_{n=1}^{\infty} \frac{1}{n^{0.3}}$?

    p-series. Has form $\sum \frac{1}{n^p}$ with $p = 0.3$.

26. Determine the convergence of $\sum_{n=1}^{\infty} \frac{1}{n^{4}}$.

    Converges. Since $p = 4 > 1$, this p-series converges.

27. What is the third term of a harmonic series?

    $\frac{1}{3}$. The third term in the harmonic series is $\frac{1}{3}$.

28. State the convergence of $\sum_{n=1}^{\infty} \frac{1}{n^{0.8}}$.

    Diverges. Since $p = 0.8 < 1$, this p-series diverges.

29. Identify if $\sum_{n=1}^{\infty} \frac{1}{n^{1/2}}$ converges or diverges.

    Diverges. Since $p = 0.5 < 1$, this p-series diverges.

30. What is the first term of a harmonic series?

    1. The harmonic series starts with $\frac{1}{1} = 1$.

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