Final answer:
The harmonic series diverges by examining its partial sums. It is proven by showing that as you sum the series, the totals exceed any given number. This concept is essential in mathematical analysis, specifically in the behavior of sequences and series.
Step-by-step explanation:
The original question regarding whether the harmonic series diverges by examining partial sums is a problem in the realm of mathematics, specifically within the topic of series and sequences. The harmonic series is defined as the sum of 1/n for n = 1 to infinity. The divergence of this series is a classical result in mathematical analysis.
To show this, one would consider the n-th partial sum of the harmonic series, Sn = 1 + 1/2 + 1/3 + ... + 1/n. By grouping and comparing these terms to halves, it can be seen that each group (after the first) is greater than 1/2, thus showing that as n increases, the partial sums exceed any given number, which proves that the series diverges.
This approach avoids the integral test, relying instead on the comparison of partial sums. The concept of divergence and convergence is crucial in the evaluation of series, much like unlimited sequences can approach a finite limit or extend without bound. The harmonic series is an example where partial sums grow without bound, indicating divergence.
Other True/False Questions
- Amplitudes of waves indeed add up only if they are propagating in the same line, so the statement is True.
- Waves can superimpose even if their frequencies are different; thus that statement is True.
- The use of the Pythagorean theorem is correct in the context of vectors at right angles to each other, making that statement True.
- There are indeed two types of interference: constructive and destructive. Hence, this statement is True.
- Electric-field lines from a positive point charge do spread out radially and point outward, so that statement is True.
- The observed frequency does not become infinite when the source is moving at the speed of sound; that statement is False.