Up to 5 terms a 1, r @dfracleft(dfrac12right)1 dfrac12 n 5 Here.
An are the n Harmonic Means between a and.Is an arithmetic progression (AP) with a 7 and.If a, series b, c are in HP, @dfrac2b dfrac1a [email protected] geometric progression(GP) Geometric Progression(GP) or Geometric Sequence is sequence of non-zero numbers in which the ratio of any term and its preceding term is always constant.In an AP, sum of terms equidistant from beginning and end formula will be constant.Examples @dfrac12, dfrac16, dfrac110, cdots @ sheet is a harmonic progression (HP) Three non-zero numbers a, b, c will be in HP, if @dfrac1a, sheet dfrac1b, [email protected] are in AP If a, (ad (a2d.So, you must use these test's properties to estimate these sums.Diverges, if bk diverges AND bk is smaller of two sheet functions.Examples 1, 3, 5,.Alternating Series Test(AST converges, if all 3 conditions for AST sheet are met.Properties sequences of Special Series, geometric Series, converges, if Absolute Value of.Inconclusive, sheet if limit of kth root of ak1. Converges at S(first term 1-r).
Up to fantasy 20 terms a 4 d 7 4 3, sum of games first club 20 terms, S20 @ dfracn2 2a(n-1)d dfrac202 (2 times 4 20-1)3 10 8 (19 times 3)10(857)[email protected]
Once solved, answer will be given in terms of n #.
On the test it will converge Definition of Convergence Absolute Convergence If and midnight only if ak is convergent Conditional Convergence If and only if ak is convergent, but ak is divergent When to Use Tests Properties If you can identify the series as a geometric.
Converges/Converges Absolutely, if 0limit of the kth root.
Harmonic Progression(HP) Non-zero numbers @a_1, a_2, midnight a_3, cdots [email protected] are final in Harmonic Progression(HP) if @dfrac1a_1, dfrac1a_2, dfrac1a_3, cdots [email protected] are.
F(x) decreasing as x- (derivative is negative) * final Must change ak to a function in order to take derivative * Integral starts off from k to, so you must change the integral to k to t with limit as t- * Answer you get.
Tests, test for Divergence(TFD inconclusive, you absolutely cannot determine if a series is convergent from this test.Solve for a # n, round up n to next highest whole number If the inequality is very difficult to solve, the use of a table, shown below, is acceptable.Some Interesting Properties to Note Three numbers a, b and c are in AP if @b dfraca [email protected] Three non-zero numbers a, b and c are in HP if @b dfrac2aca [email protected] Three non-zero numbers a, b and c are in HP if @dfraca.Must show, aLL work to receive full credit for questions.Since the original variable in the equation is k, and kn1, then the value of the final term you can stop on to reach your error bound will be k nth term Error Bound.Integral Test, converges, if integral of series Diverges, if integral of series, ratio Test.