Algebra and Trigonometry Quiz

The natural logarithm of a number is the exponent to which the mathematical constant e must be raised to equal that number. It is represented as ln(x).

The common logarithm of a number is the exponent to which 10 must be raised to equal that number. It is represented as log(x) or log10(x).

Different bases lead to different shapes of logarithmic curves. The logarithmic function can be shifted horizontally by changing the base. Larger base values result in slower growth of the function.

The exponential and logarithmic functions are inverses of each other. This means that if you apply one function, and then the other, you get the original number back.

A function is strictly increasing if its output value always becomes larger as the input value increases. A function is strictly decreasing if its output value always becomes smaller as the input value increases.

For any positive number b and any positive number x, the expression b raised to the power of the logarithm base b of x equals x (b^logb(x) = x). This rule helps solve logarithmic equations.

The logarithm of a product is equal to the sum of the logarithms of the individual factors. (logb(x*y) = logb(x) + logb(y)).

The logarithm of a power is equal to the exponent multiplied by the logarithm of the base. (logb(x^y) = y*logb(x)).

A function defined as the ratio of sine to cosine, represented by tan(x) = sin(x) / cos(x), where x is an angle in radians.

The trigonometric function tangent is undefined for values of x where cos(x) = 0, which occurs at x = π/2 + nπ, where n is any integer.

The tangent function has a period of π, meaning its values repeat every π radians. This implies tan(x + 2πn) = tan(x), for any integer n.

The trigonometric function cotangent is the reciprocal of the tangent function, defined as cot(x) = cos(x) / sin(x). It's undefined where sin(x) = 0, which occurs at x = nπ, where n is any integer.

A fundamental trigonometric identity that relates sine and cosine: sin²(x) + cos²(x) = 1.

The cosine of an angle x is equal to the sine of the angle shifted by π/2: cos(x) = sin(x + π/2).

The sine and cosine functions have a period of 2π: sin(x + 2π) = sin(x) and cos(x + 2π) = cos(x).

An ordered list of numbers, often infinite, where each number is associated with a natural number (index), denoted as (x1, x2, x3, ...).

A sequence (an) tends to infinity if, for any number M, there exists an N such that for all n greater than N, an is greater than M. This is denoted as lim (an) = ∞ or an → ∞.

A sequence (an) tends to minus infinity if, for any number M, there exists an N such that for all n greater than N, an is less than M. This is denoted as lim (an) = -∞ or an → -∞.

The limit of a sum of sequences (an+bn) is equal to the sum of their individual limits, as long as both sequences have finite limits. This means that the limit of the combined sequence is simply the sum of the limits of the two individual sequences.

A sequence where the terms get progressively closer to a specific value as 'n' increases.

A sequence where the distance between its terms and a value 'x' becomes arbitrarily small as 'n' increases.

The property of a sequence where the terms do not approach any specific value as 'n' increases.

For any arbitrarily small 'epsilon' value, there exists a natural number 'N' such that for all 'n' greater than 'N', the difference between the 'n'th term of the sequence and the limit 'x' is smaller then 'epsilon.'

A sequence that approaches a specific value as the 'n'th term increases, but never reaches it.

A sequence where the terms get progressively smaller as 'n' increases.

A sequence where its terms increase as 'n' increases, but it may still approach a specific limit.

The behavior of a sequence at 'infinity' can't be fully described by just looking at its first few terms. It needs a proper definition.

Given two sequences (an) and (bn) with finite limits a and b respectively, the limit of their sum (an + bn) is equal to the sum of their individual limits (a + b).

For any arbitrary positive value (epsilon), there exists a natural number (N) such that for all values of n greater than N, the absolute difference between the nth term of the sequence (an) and its limit (a) is less than epsilon.

A sequence converges to a specific value if its terms get arbitrarily close to that value as 'n' (the index of the sequence) increases.

The product of two sequences with finite limits also has a finite limit, and this limit is equal to the product of the limits of the individual sequences.

Given two sequences (an) and (bn) where both have finite limits and the limit of (bn) is not zero, the limit of their ratio (an/bn) is equal to the ratio of their individual limits.

A sequence converges to a specific value if the terms in the sequence get closer and closer to that value as 'n' (the index of the sequence) increases.

The statement that lim (an · bn) = lim an · lim bn when both (an) and (bn) have finite limits.

The statement that lim (an/bn) = (lim an) / (lim bn) when both (an) and (bn) have finite limits and lim bn ≠ 0.

For a sequence 'a', the infinite sum is defined as the limit of the sum of the first 'n' terms of the sequence as 'n' approaches infinity. The limit must exist for the infinite sum to be defined.

An infinite sum exists when the limit of the sum of the first 'n' terms as 'n' approaches infinity exists. This limit is the value of the infinite sum.

In an infinite sum, the sequence of partial sums (sums of the first 'n' terms) converges to a limit as 'n' approaches infinity.

The infinite sum of a geometric sequence with a common ratio 'q' (where the absolute value of 'q' is less than 1) is given by the formula a1/(1-q), where a1 is the first term.

The limit of a sequence can be calculated by interchanging summation and multiplication with taking the limits (using the Theorem of Limit Properties).

The limit of a sequence can be calculated by using the properties of limits to simplify the expression. This often involves dividing both the numerator and denominator by the highest power of 'n' in the expression.

The limit of a sequence can be calculated by using the Squeeze Theorem. This theorem states that if two sequences converge to the same limit and a third sequence is bounded between them, then the third sequence also converges to that limit.

The limit of a sequence can be calculated by using the Monotone Convergence Theorem. This theorem states that if a sequence is bounded and monotone (either increasing or decreasing), then it converges to a limit.