Series & Sequences Solver

The Series & Sequences Solver answers the standard questions about sequences and infinite series. Does the sequence converge, and to what. Does the series converge, absolutely or conditionally, by which test. What is the sum of a geometric or telescoping series. What is the Taylor or Maclaurin polynomial of a function to a given degree, and what is its radius and interval of convergence. For a convergence question it picks the test that applies, states its hypotheses, checks them explicitly, and draws the conclusion. Power series manipulations, differentiating or integrating a known series term by term to obtain a new one, are shown with the index shift written out.

Type the series with its index and starting value, or photograph it: "sum from n=2 to infinity of 1/(n ln n)", "Maclaurin series of cos x to x^6", "radius of convergence of sum (x-1)^n / n 2^n". If the assignment requires a particular test, name it: "use the ratio test". State the degree you need for a polynomial approximation and the center for a Taylor series. Set Answer form to exact for closed-form sums and decimal for numeric approximations. For an error-bound question, say which bound the course uses, alternating series or Taylor's inequality, and the solver computes it with the term or derivative bound identified. For a sequence, give the general term or the first few terms with the recurrence.

Output: the conclusion first, converges or diverges, the sum, or the expansion. Then the test with its conditions checked one at a time, or the term-by-term derivation of the expansion. Interval of convergence problems show the ratio test, then the endpoint checks separately. The working is shown so you can check it. For a single limit of a sequence term, the Limit Calculator is faster. For the integral in an integral test, the Integral Calculator shows that step in more detail. A follow-up can rerun the convergence question with a different test, which is useful when the assignment asks you to compare two, or extend a Taylor polynomial to a higher degree. Partial sums are tabulated on request to show the numeric approach to the limit.