A double sum of series is the sum of the terms of a series where each term is itself a sum of two variables or indices. Mathematically, it can be represented as the sum of the sum of aij, where i and j are the indices of the terms being summed. Double sums often arise in the context of multivariable calculus, where they are used to represent the sum of a function over a two-dimensional region or domain. Double sums can also be used to represent the sum of a series of series, where each term of the outer series is itself a series. **

The sum of the series nn1 is infinite. This series diverges, meaning it does not have a finite sum. As n approaches infinity, the terms of the series also approach infinity, so the sum of the series is not a finite value. Therefore, the sum of the series nn1 is undefined. **

The sum formula for the harmonic series is given by the equation: 1 + 1/2 + 1/3 + 1/4 + ... + 1/n = ln(n) + γ, where ln(n) is the natural logarithm of n and γ is the Euler-Mascheroni constant. This formula shows that the sum of the harmonic series diverges as n approaches infinity. **

To calculate the lower sum of a function over a given interval, you divide the interval into subintervals and then find the minimum value of the function within each subinterval. You then multiply each minimum value by the width of its corresponding subinterval and sum these products to obtain the lower sum. To calculate the upper sum, you do the same process, but instead of finding the minimum value of the function within each subinterval, you find the maximum value. Then you multiply each maximum value by the width of its corresponding subinterval and sum these products to obtain the upper sum. **

The sum series expansion of the Pochhammer symbol, also known as the rising factorial, is given by: (x)_n = x(x+1)(x+2)...(x+n-1) = Σ(k=0 to n) (n choose k) * x^k This expansion represents the product of consecutive terms starting from x and increasing by 1, up to n terms. It can also be expressed as a sum of binomial coefficients multiplied by powers of x. **

The series sum 1/n^k converges to a finite value when k is greater than 1. Specifically, it converges to a value of 1/(k-1) when k is greater than 1. This is known as the p-series and is a well-known result in calculus. When k is less than or equal to 1, the series diverges. **

Sum terms are individual components of a mathematical expression that are added together to form the total sum. In an algebraic expression, sum terms are separated by plus or minus signs and can include variables, constants, and coefficients. For example, in the expression 3x + 2y - 5, the terms 3x, 2y, and -5 are sum terms. Understanding sum terms is important for simplifying and solving algebraic equations. **

To calculate the upper sum and lower sum of the function 15x^3 over a given interval, you would first need to divide the interval into subintervals of equal width. Then, you would evaluate the function at the right endpoint of each subinterval to find the upper sum, and at the left endpoint of each subinterval to find the lower sum. Finally, you would sum up the values obtained for each subinterval to get the total upper sum and lower sum of the function over the interval. **