Non-zero constant.sutupidmath said:yeah that is right, since the harmonic series diverges, it also diverges when we multiply it by a constant.
Kummer said:Non-zero constant.
Not to offend you but that is what I like to call a "physics-type mistake".
The comparison test is a method used to determine the convergence or divergence of a series by comparing it to another series whose convergence or divergence is already known. It states that if the terms of the series in question are always less than or equal to the terms of a convergent series, then the original series must also converge. Similarly, if the terms are always greater than or equal to the terms of a divergent series, then the original series must also diverge.
The direct comparison test compares the terms of the series in question directly to the terms of a known convergent or divergent series, while the limit comparison test compares the ratio of the terms of the two series as the index approaches infinity. The direct comparison test is typically easier to use, but the limit comparison test can be useful when the terms of the series do not have a clear comparison.
No, the comparison test only tells us whether a series converges or diverges. To determine the exact value of a convergent series, other methods such as the geometric series test or the integral test must be used.
Yes, the comparison test can only be used when the terms of the series are always positive. It also cannot be used to determine the convergence or divergence of alternating series.
No, the comparison test can only be used for series with positive terms. However, the absolute convergence test can be used to determine the convergence of series with both positive and negative terms.