Never underestimate the binomial theorem even if you just want an inequality. As an example, look at theorem 3. This inequality allows us to decide how much 'weight' we want to give to a particular term in a product.
This can be used to show that the product of Riemann integrable functions is still Riemann integrable. The Weierstrass M-test is the first friend you call when dealing with series of functions. Ask yourself whether the problem you're working on can be generalized to topology first. Think about compactness and connectedness and the abstract theorems about them you already know. It's not a complicated technique but certainly a recurring one in analysis.
Some of the methods that I have collected under analysis include: Collect inequalities. Collect algebraic identities. Collect limits even slowly converging ones in their various forms. Exploit density, compactness, and connectedness. Attempt to seesaw the quantities you're working with. Give yourself some buffer room. Introduce a variable, even if it makes your problem seemingly more difficult. Take the Herglotz point-of-view. Be optimistic. Thank you! Always impressed by your answers as well.
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Linear Relations and Functions. Simplifying Complex Fractions. Writing Algebraic Expressions. Factoring General Polynomials. The Slope of a Line. Positive and Negative Slopes. Solving Linear Inequalities with Fractions. Solving Linear Inequalities. Factoring Trinomials. Equations Quadratic in Form. Negative Integral Exponents. Solving Equations with Variables on Each Side.
Dividing a Polynomial by a Binomial. Synthetic Division. Combining Operations. Linear Equations. Multiplying Fractions. Dividing Monomials. Multiplication Property of Equality. Dividing Complex Numbers. Solving Absolute Value Equations. Dividing Rational Expressions. Solving Quadratic Equations. The Product and Quotient Rules.
Linear Systems of Equations with No Solution. Solving Quadratic Equations by Completing the Square. Sample Problem. Find GCF. Find LCM. The resulting loud thud usually confirms for students that an infinite sum can indeed have a finite answer. For further reading, see my series on arithmetic and geometric series.
Sometimes I teach my students how people converted decimal expansions into fractions before there was a button on a calculator to do this for them. For example, to convert into a fraction, the first step from the Bag of Tricks is to multiply by How do we change this into a decimal? Notice that the decimal parts of both and are the same.
Subtracting, the decimal parts cancel, leaving. In my experience, most students — even senior math majors who have taken a few theorem-proof classes and hence are no dummies — are a little stunned when they see this procedure for the first time. Pop out your calculators. Then punch in 16 divided by Indeed, my experience many students really do need this technological confirmation to be psychologically sure that it really did work.
See also my fuller post on this topic as well as the index for the entire series. For example, to find the antiderivative of , the first step is far from obvious:. I became clairvoyant. The joke, of course, is that the only reason that I multiplied by is that someone figured out that multiplying by at this juncture would actually be helpful. Subtracting from , the decimal parts cancel, leaving. I learned this procedure when I was very young; however, in modern times, this procedure appears to be a dying art.
Every once in a while, students encounter a step that seems far from obvious. To give one example, to evaluate the series. However, students may wonder how they were ever supposed to think of the first step for themselves.
Socrates gave the Bag of Tricks to Plato, Plato gave it to Aristotle, it passed down the generations, my teacher taught the Bag of Tricks to me, and I teach it to my students. Math Pages has a nice discussion about mathematical aspects of this problem, including connections to the Laws of Sines and Cosines and to various trig identities.
I have a story that I tell my students about the patented Bag of Tricks: Socrates gave the Bag of Tricks to Plato, Plato gave it to Aristotle, it passed down the generations, my teacher taught the Bag of Tricks to me, and I teach it to my students.
I recommend this problem for advanced geometry students who need to be challenged; even bright students will be stumped concerning coming up with the requisite trick on their own. Indeed, the problem still remains quite challenging even after the trick is shown. I conclude this series of posts by considering the formula for an infinite geometric series.
Somewhat surprisingly to students , the formula for an infinite geometric series is actually easier to remember than the formula for a finite geometric series. One way of deriving the formula parallels the derivation for a finite geometric series.
If are the first terms of an infinite geometric sequence, let. Once again, we multiply both sides by. Next, we add the two equations. Notice that almost everything cancels on the right-hand side… except for the leading term. This is the largest value minus the smallest value. The function accepts multiple ranges to check the number against, and returns true if it is within the bounds of at least one of them. If speed is meters per minute and distance is 10 meters , then the duration will be 10 minutes.
Add an array of numbers together. Count the number of decimals in a number. Get an array of angles sorted by which is closest to the target. Get the median from an array of numbers. Get the all the distances between multiple sets of coordinates sorted from shortest to longest. Get the distance between two sets of coordinates. Get a point between two sets of coordinates. Convert a single number or an array with two numbers to a set of coordinates.
Get the angle from one set of coordinates to another. Check if an angle is within one or multiple ranges of angles. Get the boundaries from an array of coordinates.
Check if a set of coordinates is inside a radius from the origin 0,0. Get the slope of a line also called a gradient. Get circumference perimeter of a circle from its radius. Get surface area of a circle from its radius. Get an array of equally distanced points on the perimeter of a circle. Get surface area of a cone. Get volume of a cone.
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