Calculus: Riemann Sums, Bounding Integrals, and Convergence Tests

When integrating the function $f(x) = \frac{1}{x^p}$ on the interval from $2$ to $3$ , the value is bounded by $\frac{1}{3^p}$ .
This specific bounding technique depends heavily on the interval of integration. While it holds for $[2, 3]$, it would not work the same way if the integration occurred between $1$ and $2$ .
This process effectively approximates the integral using a single Riemann sum, where the height of the rectangle is determined by the function's behavior on that interval.

The behavior of the function (decreasing vs. increasing) determines how the integral bounds the series.
Decreasing Functions: When a function is strictly decreasing, the area under the curve can be compared to the series using upper and lower bounds. A "red area" (the integral) can be viewed as being approximately the same as the sum, with a "white area" representing the discrepancy between the rectangular sum and the continuous integral.
Increasing Functions: If a function were increasing (e.g., following a sequence like $2, 3, 4...$ ), a lower bound for the term $f(3)$ would be the integral from $2$ to $3$ of $f(x) dx$ .
General Range for Bounding: The range for these approximations typically involves intervals such as $m-1$ to $n$ .
When writing out terms for a sum starting from $k=1$ , it is observable that the first term is bounded by the corresponding integral, as is the second term, even if the overall representation is not continuous.

The Comparison Test is established using two sequences, $a_n$ and $b_n$ .
The condition for the test is defined as: $0 \leq a_n \leq b_n$
If the larger sequence $b_n$ converges, then the smaller sequence $a_n$ must also converge.
Conversely, if the smaller sequence diverges to infinity, the larger one must also diverge.
This logic applies to sequences and series in the same way it applies to improper integrals.

Consider the sequence: $\frac{2 \pm (-1)^n}{n^p}$
To determine when this converges, one can use comparison methods or limit comparisons.
The limit comparison might involve comparing the term to $\frac{1}{n^p}$ or examining the behavior of related functions like $\frac{1}{x + 0.5}$ .
In some cases, the sequence can simply be "bounded" directly to determine convergence behavior.

The instructor discourages the use of "mnemonics" or rote memorization regarding the Integral Test (e.g., memorizing simply that "if the integral converges, the series converges").
Visual Logic: Instead of memorization, students are encouraged to "write things off" or draw the areas to see which is the upper bound and which is the lower bound.
First Term Isolation: In practical application, it is often necessary to "pick out the first term" of the series separately before applying the integral test to the remaining tail of the sequence.

Question: Are there cases where you have to bound it specifically?
Answer: Yes, for example with expressions like $\frac{2 + (-1)^n}{n^p}$ , you have to decide when it converges by comparing it to known p-series or using the limit comparison test where you look at the growth rate.
Discussion on Quiz: The instructor confirmed that there will be a quiz tomorrow because the material was repeated.