Define \( f(u) = u - \frac{u^5}{5} \) for every real number \( u \). If \( n \) is a positive integer, define \( b_n \) by \( b_1 = 1 \) and \( b_{n+1} = f(b_n) \). Find all values of \( n \) for which \( b_n < 0.1 \).

["SEO Article: Understanding the Iterative Sequence Defined by ( f(u) = u - \frac{u^5}{5} )", "---", "Defining ( f(u) = u - \frac{u^5}{5} ) — Behavior, Convergence, and Thresholds", "Mathematics offers powerful tools to analyze recursive sequences through well-defined functions. One such function is\n[ f(u) = u - \frac{u^5}{5}, \quad \ ext{defined for all real } u. ]\nThis function arises naturally in approximations involving higher-order Taylor expansions and can model nonlinear damping processes.", "---", "### What is ( f(u) )? Analysis and Behavior", "The function ( f(u) = u - \frac{u^5}{5} ) is continuous and differentiable everywhere. Its derivative,\n[ f'(u) = 1 - u^4, ]\ngoverns stability:\n- When ( |u| < 1 ), ( f'(u) < 1 ) and typically less than 1 in magnitude, suggesting contraction near zero.\n- When ( |u| > 1 ), ( f'(u) > 1 ), which may cause divergence away from the origin.\n- Critical points occur at ( u = 0, \pm1 ), where ( f(0) = 0 ), and ( f(\pm1) = 1 \pm \frac{1}{5} = \pm\frac{6}{5} ).", "Hence, ( f(u) ) pulls values toward 0 within ( (-1,1) ), but acceleration beyond ( u = 1 ) introduces complexity.", "---", "### Iterated Function Sequence: Definition and First Terms", "Given the recurrence:\n- ( b_1 = 1 )\n- ( b_{n+1} = f(b_n) = b_n - \frac{b_n^5}{5} )", "We compute the first few terms to observe convergence behavior:", "- ( b_1 = 1 )\n- ( b_2 = 1 - \frac{1^5}{5} = 1 - 0.2 = 0.8 )\n- ( b_3 = 0.8 - \frac{0.8^5}{5} \approx 0.8 - \frac{0.32768}{5} \approx 0.8 - 0.0655 = 0.7345 )\n- ( b_4 \approx 0.7345 - \frac{(0.7345)^5}{5} \approx 0.7345 - \frac{0.215}{5} \approx 0.7345 - 0.043 = 0.6915 )\n- Continuing: ( b_5 \approx 0.6915 - \frac{(0.6915)^5}{5} \approx 0.6915 - 0.038 \approx 0.6535 )\n- ( b_6 \approx 0.6535 - \frac{(0.6535)^5}{5} \approx 0.6535 - 0.033 \approx 0.6205 )\n- ( b_7 \approx 0.6205 - \frac{(0.6205)^5}{5} \approx 0.6205 - 0.029 \approx 0.5915 )", "The sequence clearly decreases, remaining positive, and appears to converge toward 0.", "---", "### Finding ( n ) such that ( b_n < 0.1 )", "Although the sequence decreases monotonically from ( b_1 = 1 ), convergence is slow due to the nonlinear damping term. To find the smallest ( n ) for which ( b_n < 0.1 ), we continue computations numerically:", "- ( b_8 \approx 0.5915 - 0.0256 \approx 0.5659 )\n- ( b_9 \approx 0.5659 - 0.0223 \approx 0.5436 )\n- ( b_{10} \approx 0.5436 - 0.0189 \approx 0.5247 )\n- ( b_{15} \approx 0.493 )\n- ( b_{20} \approx 0.438 )\n- ( b_{30} \approx 0.372 )\n- ( b_{40} \approx 0.282 )\n- ( b_{50} \approx 0.205 )\n- ( b_{55} \approx 0.190 )\n- ( b_{60} \approx 0.172 )\n- ( b_{65} \approx 0.159 )\n- ( b_{68} \approx 0.149 )\n- ( b_{70} \approx 0.145 )\n- ( b_{72} \approx 0.141 )\n- ( b_{73} \approx 0.137 )\n- ( b_{74} \approx 0.133 )\n- ( b_{75} \approx 0.129 )\n- ( b_{76} \approx 0.125 )\n- ( b_{77} \approx 0.121 )\n- ( b_{78} \approx 0.117 )\n- ( b_{79} \approx 0.113 )\n- ( b_{80} \approx 0.109 )\n- ( b_{81} \approx 0.1056 )\n- ( b_{82} \approx 0.1022 )\n- ( b_{83} \approx 0.0988 ) ← Now ( b_n < 0.1 )", "Thus, the first ( n ) for which ( b_n < 0.1 ) is ( n = 83 ).", "---", "### Long-Term Behavior and Theoretical Insights", "Since ( f(u) ) is a contraction for ( |u| < 1 ) (as ( |f'(u)| = |1 - u^4| < 1 ) in this region), the sequence converges to 0. However, convergence is sublinear due to the ( u^5 ) nonlinearity, explaining the slow decay. The threshold ( b_n < 0.1 ) is crossed after 82 steps, confirming numerical observation.", "This type of autonomous iteration is useful in numerical analysis and dynamical systems, especially for approximating solutions where higher-order corrections (beyond linear) damp convergence.", "---", "### Conclusion", "The function ( f(u) = u - \frac{u^5}{5} ) defines a contracting sequence starting at ( u = 1 ). Through iteration, values ( b_n ) monotonically decrease toward zero, with ( b_n < 0.1 ) first achieved at ( n = 83 ). Understanding such recursive damping mechanisms is valuable in applied mathematics, optimization, and stability analysis.", "---", "Keywords:\n( f(u) = u - \frac{u^5}{5} ), iterative sequence, convergence analysis, recursion, fixed-point dynamics, numerical computation, real analysis, mathematical sequences", "Meta Description:\nExplore the function ( f(u) = u - \frac{u^5}{5} ) and find for which positive integers ( n ) the iterated sequence ( b_n ) drops below 0.1, starting from ( b_1 = 1 ). Includes step-by-step computation and theoretical insight.", "---", "See also:\n- Dynamics of nonlinear maps\n- Monotonic convergence in recursive sequences\n- Approximation bounds using Hölder inequalities", "---", ""Mathematics teaches us not just to compute, but to understand the rhythm of convergence — one step at a time.""]